36:I[651451,["/_next/static/chunks/02and8gctjrt2.js?dpl=nogit-1790362117","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362117","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362117","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362117","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362117","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362117","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362117","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362117","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362117","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362117","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362117","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362117","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362117","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362117","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362117","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362117","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362117","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362117","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362117","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362117","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362117","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362117","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362117","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362117","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362117","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362117","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362117","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362117","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362117","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362117","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362117","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362117","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362117","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362117","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362117","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362117","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362117","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362117","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362117","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362117","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362117","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362117","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362117","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362117","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362117","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362117","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362117","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362117","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362117","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362117","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362117","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362117","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362117","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362117","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362117","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362117","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362117","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362117","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362117","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362117","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362117","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362117","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362117","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362117","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362117","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362117","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362117","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362117","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362117","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362117"],"UserProgressProvider"] 37:I[196793,["/_next/static/chunks/02and8gctjrt2.js?dpl=nogit-1790362117","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362117","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362117","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362117","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362117","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362117","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362117","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362117","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362117","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362117","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362117","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362117","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362117","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362117","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362117","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362117","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362117","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362117","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362117","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362117","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362117","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362117","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362117","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362117","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362117","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362117","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362117","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362117","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362117","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362117","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362117","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362117","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362117","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362117","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362117","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362117","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362117","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362117","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362117","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362117","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362117","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362117","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362117","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362117","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362117","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362117","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362117","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362117","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362117","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362117","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362117","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362117","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362117","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362117","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362117","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362117","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362117","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362117","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362117","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362117","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362117","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362117","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362117","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362117","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362117","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362117","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362117","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362117","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362117","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362117"],"Gingerbread"] 39:I[716297,["/_next/static/chunks/02and8gctjrt2.js?dpl=nogit-1790362117","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362117","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362117","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362117","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362117","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362117","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362117","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362117","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362117","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362117","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362117","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362117","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362117","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362117","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362117","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362117","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362117","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362117","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362117","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362117","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362117","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362117","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362117","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362117","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362117","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362117","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362117","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362117","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362117","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362117","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362117","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362117","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362117","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362117","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362117","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362117","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362117","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362117","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362117","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362117","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362117","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362117","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362117","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362117","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362117","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362117","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362117","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362117","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362117","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362117","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362117","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362117","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362117","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362117","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362117","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362117","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362117","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362117","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362117","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362117","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362117","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362117","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362117","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362117","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362117","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362117","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362117","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362117","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362117","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362117"],"TopicRecentlyViewed"] 3a:I[647935,["/_next/static/chunks/02and8gctjrt2.js?dpl=nogit-1790362117","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362117","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362117","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362117","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362117","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362117","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362117","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362117","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362117","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362117","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362117","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362117","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362117","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362117","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362117","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362117","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362117","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362117","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362117","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362117","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362117","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362117","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362117","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362117","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362117","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362117","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362117","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362117","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362117","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362117","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362117","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362117","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362117","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362117","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362117","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362117","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362117","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362117","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362117","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362117","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362117","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362117","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362117","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362117","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362117","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362117","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362117","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362117","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362117","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362117","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362117","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362117","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362117","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362117","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362117","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362117","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362117","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362117","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362117","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362117","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362117","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362117","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362117","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362117","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362117","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362117","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362117","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362117","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362117","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362117"],"RecordResourceResume"] 3b:I[426249,["/_next/static/chunks/02and8gctjrt2.js?dpl=nogit-1790362117","/_next/static/chunks/2mu6-wo6ccpi3.js?dpl=nogit-1790362117","/_next/static/chunks/2epfhfe5im7r6.js?dpl=nogit-1790362117","/_next/static/chunks/1ew75_i487l-5.js?dpl=nogit-1790362117","/_next/static/chunks/0p4rexedanx97.js?dpl=nogit-1790362117","/_next/static/chunks/0u008gt6051u7.js?dpl=nogit-1790362117","/_next/static/chunks/1odwrcl12kfnl.js?dpl=nogit-1790362117","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1790362117","/_next/static/chunks/3lf9kdvvybemj.js?dpl=nogit-1790362117","/_next/static/chunks/2hyw6fyf3iqeh.js?dpl=nogit-1790362117","/_next/static/chunks/03q735zs9dbpu.js?dpl=nogit-1790362117","/_next/static/chunks/3r5hh4t0jf4es.js?dpl=nogit-1790362117","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1790362117","/_next/static/chunks/1h4wccj42vygd.js?dpl=nogit-1790362117","/_next/static/chunks/12xoy6h_w_le6.js?dpl=nogit-1790362117","/_next/static/chunks/2os7nxouq91eq.js?dpl=nogit-1790362117","/_next/static/chunks/3ht89vb0p09bm.js?dpl=nogit-1790362117","/_next/static/chunks/3ocoqwe4aht_u.js?dpl=nogit-1790362117","/_next/static/chunks/32qmkt01ta8b2.js?dpl=nogit-1790362117","/_next/static/chunks/3jpl-42mbad5n.js?dpl=nogit-1790362117","/_next/static/chunks/3ao1re0xmmur_.js?dpl=nogit-1790362117","/_next/static/chunks/3xa2z7fbv2riw.js?dpl=nogit-1790362117","/_next/static/chunks/14kf-an-d2n-e.js?dpl=nogit-1790362117","/_next/static/chunks/381o-754ev9ia.js?dpl=nogit-1790362117","/_next/static/chunks/2tr2cll18vqv3.js?dpl=nogit-1790362117","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1790362117","/_next/static/chunks/3kom3ps_bn2q6.js?dpl=nogit-1790362117","/_next/static/chunks/1x_yxljsdq0vl.js?dpl=nogit-1790362117","/_next/static/chunks/1xkpwfqqdrdio.js?dpl=nogit-1790362117","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1790362117","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1790362117","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1790362117","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1790362117","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1790362117","/_next/static/chunks/440m38krgocdi.js?dpl=nogit-1790362117","/_next/static/chunks/2p6gbbs6zvfpy.js?dpl=nogit-1790362117","/_next/static/chunks/172wsar4r7zzx.js?dpl=nogit-1790362117","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1790362117","/_next/static/chunks/1rbw_-uux6mpq.js?dpl=nogit-1790362117","/_next/static/chunks/1s6j_7_p0ugo9.js?dpl=nogit-1790362117","/_next/static/chunks/34ux0fo6f-rm8.js?dpl=nogit-1790362117","/_next/static/chunks/1pp_uf_jx5j9r.js?dpl=nogit-1790362117","/_next/static/chunks/2b6r02yb2zswo.js?dpl=nogit-1790362117","/_next/static/chunks/422ba61wql3d7.js?dpl=nogit-1790362117","/_next/static/chunks/2vg15qyptj-oc.js?dpl=nogit-1790362117","/_next/static/chunks/0d2u5k628p_lw.js?dpl=nogit-1790362117","/_next/static/chunks/3jzaxe43n3e1e.js?dpl=nogit-1790362117","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1790362117","/_next/static/chunks/02e_f03j5ij1l.js?dpl=nogit-1790362117","/_next/static/chunks/12mv6p8rwcl5o.js?dpl=nogit-1790362117","/_next/static/chunks/0v5zhouho0zlq.js?dpl=nogit-1790362117","/_next/static/chunks/41x40gd5jiq0g.js?dpl=nogit-1790362117","/_next/static/chunks/2nnmvgmq4euk9.js?dpl=nogit-1790362117","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1790362117","/_next/static/chunks/3wqp-h5xnebw9.js?dpl=nogit-1790362117","/_next/static/chunks/30dopbgd1dkf9.js?dpl=nogit-1790362117","/_next/static/chunks/22z8pa0ys3w4l.js?dpl=nogit-1790362117","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1790362117","/_next/static/chunks/2ywf8cxod8dkv.js?dpl=nogit-1790362117","/_next/static/chunks/43ajf4ofaeu5y.js?dpl=nogit-1790362117","/_next/static/chunks/1__39-dgg7p8-.js?dpl=nogit-1790362117","/_next/static/chunks/1zt4k83kx3bkw.js?dpl=nogit-1790362117","/_next/static/chunks/0rt2_go1it43r.js?dpl=nogit-1790362117","/_next/static/chunks/200x0be4hbjpj.js?dpl=nogit-1790362117","/_next/static/chunks/35bn8r0ao86_s.js?dpl=nogit-1790362117","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1790362117","/_next/static/chunks/2wlqtkfegn9hx.js?dpl=nogit-1790362117","/_next/static/chunks/2y341-r8ncizo.js?dpl=nogit-1790362117","/_next/static/chunks/03ai9i_h50rnc.js?dpl=nogit-1790362117","/_next/static/chunks/0pdzmybpnr872.js?dpl=nogit-1790362117"],"TopicQuestionBank"] 24:["$","$L36",null,{"bundle":{"topicIds":["b00a295e-cd21-4677-87b5-c55f0941d173"],"paperIds":[],"stageIds":[],"stageTopicIds":[],"stageAssessmentIds":[],"topicNoteIds":[]},"children":[["$","$L37",null,{"feature":"question-bank"}],"$L38",["$","$L39",null,{"subjectCode":"a_level_maths__ocr","subjectName":"Maths","topicSlug":"proof-by-contradiction","topicTitle":"1.1.4 Proof by contradiction (A-level only)","sectionTitle":null,"activity":"worksheet","resourceTitle":"1.1.4 Proof by contradiction (A-level only) Questions","resourceHref":"/a-level/maths/ocr/proof-by-contradiction/questions/q1","saveTrail":false}],["$","$L3a",null,{"href":"/a-level/maths/ocr/proof-by-contradiction/questions/q1","label":"1.1.4 Proof by contradiction (A-level only)","subjectCode":"a_level_maths__ocr","surface":"questions"}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"LearningResource\",\"name\":\"Question 13 – 1.1.4 Proof by contradiction (A-level only) - Questions\",\"description\":\"Practise 1.1.4 Proof by contradiction (A-level only) A Level Maths exam-style questions with worked solutions on MathsGenie.\",\"learningResourceType\":\"Practice questions\",\"educationalLevel\":\"A Level\",\"about\":{\"@type\":\"Thing\",\"name\":\"1.1.4 Proof by contradiction (A-level only)\"},\"url\":\"https://mathsgenie.co.uk/a-level/maths/ocr/proof-by-contradiction/questions/q13\",\"inLanguage\":\"en-GB\",\"provider\":{\"@id\":\"https://mathsgenie.co.uk/#organization\"}}"}}],["$","div",null,{"className":"min-h-0 w-full flex-1 overflow-hidden bg-background","children":["$","$L3b",null,{"backHref":"/a-level/maths/ocr/questions","basePath":"/a-level/maths/ocr/proof-by-contradiction/questions","formulaUrl":null,"initialIndex":12,"initialQuestionWindow":[{"id":"6b925623-c577-4449-a2dd-7a78f36a658f","revisionId":"6b43fb5b-f0f5-4440-a44d-4fd2355b5d30","questionNumber":null,"questionType":"LA","specification":"","difficulty":3,"difficultyBand":"hard","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"a0bd4ae7-e2d2-4ebf-8cfe-418bad7fef9f","label":"a","content":"A student named Elena makes the claim that there is a finite set of prime numbers $P = \\{p_1, p_2, p_3, \\dots, p_k\\}$ where $p_k$ is the largest prime. Use the method of proof by contradiction to show that Elena's claim is incorrect, and thus prove that there are infinitely many prime numbers.","markscheme":"- Assume that Elena's claim is correct and there is a finite number of primes $p_1, p_2, \\dots, p_k$. B1\n- Define $N$ as the product of all prime numbers in the set: $N = p_1 \\times p_2 \\times p_3 \\times \\dots \\times p_k$. M1\n- Construct the integer $M = N + 1$. M1\n- Explain that $M$ is not divisible by any of the primes $p_1, \\dots, p_k$ because dividing $M$ by any of these primes always leaves a remainder of 1. A1\n- State that $M$ must therefore either be prime itself or possess a prime factor that is not in Elena's original list. M1\n- Conclude that the initial assumption of a finite number of primes leads to a contradiction, hence there are infinitely many primes. A1","marks":6,"order":0,"markingRubric":{"totalMarks":6,"points":[{"id":"p1","marks":1,"description":"Assume the set of prime numbers is finite.","code":"B"},{"id":"p2","marks":1,"description":"Define the product of all primes in the finite list.","code":"M"},{"id":"p3","marks":1,"description":"Construct a new number equal to the product plus one.","code":"M"},{"id":"p4","marks":1,"description":"Justify that the new number is not divisible by any prime in the original list due to a remainder of 1.","code":"A"},{"id":"p5","marks":1,"description":"State the logical deduction that a new prime or new prime factor must exist.","code":"M"},{"id":"p6","marks":1,"description":"Conclude the proof by stating the contradiction and the final result.","code":"A"}],"finalAnswers":["There are infinitely many prime numbers"],"generatedAt":"2026-05-06T14:05:17.796Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"euclids_construction","label":"Standard Euclid's Construction","steps":[{"type":"text","order":0,"title":"State the initial assumption","content":"To prove that there are infinitely many prime numbers using a proof by contradiction, we must start by assuming the opposite of what we want to prove.\n\nWe assume that **Elena's claim is correct**, meaning there is a finite set of prime numbers:\n\n$$P = \\{p_1, p_2, p_3, \\dots, p_k\\}$$\n\nwhere $p_k$ is the largest prime number in existence.\n\n*Awarded 1 mark (B1) for stating this initial assumption.*"},{"type":"text","order":1,"title":"Construct a new integer","content":"Using our finite list of primes, we define $N$ as the product of all these prime numbers:\n\n$$N = p_1 \\times p_2 \\times p_3 \\times \\dots \\times p_k$$\n\nNow, we construct a new positive integer, $M$, by adding $1$ to this product:\n\n$$M = N + 1 = (p_1 \\times p_2 \\times p_3 \\times \\dots \\times p_k) + 1$$\n\n*Awarded 2 marks (M1 for defining the product $N$, and M1 for constructing $M = N+1$).*"},{"type":"text","order":2,"title":"Analyze divisibility of M","content":"Let's consider what happens when we divide $M$ by any of the prime numbers $p_i$ in our set $P$:\n\n$$\\frac{M}{p_i} = \\frac{(p_1 \\times p_2 \\times \\dots \\times p_k) + 1}{p_i} = \\frac{p_1 \\times p_2 \\times \\dots \\times p_k}{p_i} + \\frac{1}{p_i}$$\n\nSince $p_i$ is a factor of the product, the first term is an integer, leaving a remainder of $1$ from the second term. \n\nTherefore, $M$ is **not divisible** by any of the primes $p_1, p_2, \\dots, p_k$.\n\n*Awarded 1 mark (A1) for showing that dividing $M$ by any prime in the finite set leaves a remainder of 1.*"},{"type":"text","order":3,"title":"Identify the contradiction","content":"By the Fundamental Theorem of Arithmetic, every integer greater than $1$ is either a prime number itself or can be uniquely factored into a product of prime numbers.\n\nSince $M > 1$:\n- If $M$ is a prime number, then we have found a prime number that is not in our original list $P$ (since $M$ is much larger than $p_k$).\n- If $M$ is not prime, it must have a prime factor. Since none of the primes in our list $P$ divide $M$, this prime factor must be some new prime number that is not in our list $P$.\n\nIn either case, there exists a prime number that is not included in our supposedly complete list $P$.\n\n*Awarded 1 mark (M1) for explaining that $M$ must either be prime or have a prime factor not in the original list.*"},{"type":"text","order":4,"title":"State the final conclusion","content":"This directly contradicts our initial assumption that $P = \\{p_1, p_2, \\dots, p_k\\}$ contains all prime numbers.\n\nSince our assumption leads to a contradiction, the assumption must be false. Therefore, Elena's claim is incorrect, and we conclude that **there are infinitely many prime numbers**.\n\n*Awarded 1 mark (A1) for concluding the proof by identifying the contradiction and stating the final result clearly.*"}]},{"id":"factorial_construction","label":"Factorial-Based Construction","steps":[{"type":"text","order":0,"title":"State the initial assumption","content":"To prove Elena's claim is false using proof by contradiction, we must begin by assuming that her claim is true.\n\nAssume there is a finite set of prime numbers:\n\n$$P = \\{p_1, p_2, p_3, \\dots, p_k\\}$$\n\nwhere $p_k$ is the largest prime number in existence.\n\nThis earns the first mark **B1**.","highlights":[{"text":"Elena makes the claim that there is a finite set of prime numbers $P = \\{p_1, p_2, p_3, \\dots, p_k\\}$ where $p_k$ is the largest prime","location":"specification"}]},{"type":"text","order":1,"title":"Construct the integer $M$","content":"Using the **Factorial-Based Construction** approach, we define a new positive integer, $M$, by adding $1$ to the factorial of the largest prime, $p_k$:\n\n$$M = p_k! + 1$$\n\nwhere the factorial $p_k!$ is the product of all positive integers from $1$ up to $p_k$:\n\n$$p_k! = p_k \\times (p_k - 1) \\times (p_k - 2) \\times \\dots \\times 2 \\times 1$$\n\nThis step corresponds to the method marks **M1** and **M1**."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$M = \\htmlData{anchor=fact}{\\color{@sky}{p_k!}} + \\htmlData{anchor=one}{\\color{@amber}{1}}$"},{"text":"Since every prime $p_i$ in our finite set $P$ is less than or equal to $p_k$, it must be one of the integers in the product $p_k!$."},{"latex":"$$p_i \\text{ divides } \\color{@sky}{p_k!} \\implies \\frac{M}{p_i} = \\frac{\\color{@sky}{p_k!}}{p_i} + \\frac{\\color{@amber}{1}}{p_i}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"fact"},{"color":"amber","style":"text-color","anchor":"one"}],"connections":[{"to":"one","from":"fact","color":"sky","label":null,"style":"solid"}]},"order":2,"title":"Analyze divisibility of $M$","description":"Any prime $p_i$ in our assumed finite set must divide $p_k!$ because $p_i \\le p_k$. Dividing $M$ by $p_i$ will therefore always leave a remainder of $1$ (satisfying mark **A1**)."},{"type":"text","order":3,"title":"Examine the prime factors","content":"According to the Fundamental Theorem of Arithmetic, the integer $M$ must either be prime itself or it must be a product of prime factors.\n\n- If $M$ is prime, then since $M = p_k! + 1 > p_k$, we have found a prime number greater than $p_k$.\n- If $M$ is composite, it must have a prime factor, say $q$. Since $M$ is not divisible by any of the primes in our set $P$, this prime factor $q$ cannot be in $P$. Thus, $q$ must be greater than $p_k$ ($q > p_k$).\n\nIn either case, there exists a prime number greater than the assumed largest prime $p_k$ (earning mark **M1**)."},{"type":"text","order":4,"title":"Reach the contradiction","content":"Having found a prime number greater than $p_k$ directly contradicts our initial assumption that $p_k$ is the largest prime and that the set of prime numbers is finite.\n\nTherefore, our initial assumption must be false. Elena's claim is incorrect, and there are infinitely many prime numbers.\n\nThis final conclusion completes the proof and earns the final mark **A1**."}]}],"generatedAt":"2026-06-09T01:57:18.381Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"a0bd4ae7-e2d2-4ebf-8cfe-418bad7fef9f:euclids_construction","label":"Standard Euclid's Construction","steps":[{"type":"text","order":0,"title":"State the initial assumption","content":"To prove that there are infinitely many prime numbers using a proof by contradiction, we must start by assuming the opposite of what we want to prove.\n\nWe assume that **Elena's claim is correct**, meaning there is a finite set of prime numbers:\n\n$$P = \\{p_1, p_2, p_3, \\dots, p_k\\}$$\n\nwhere $p_k$ is the largest prime number in existence.\n\n*Awarded 1 mark (B1) for stating this initial assumption.*"},{"type":"text","order":1,"title":"Construct a new integer","content":"Using our finite list of primes, we define $N$ as the product of all these prime numbers:\n\n$$N = p_1 \\times p_2 \\times p_3 \\times \\dots \\times p_k$$\n\nNow, we construct a new positive integer, $M$, by adding $1$ to this product:\n\n$$M = N + 1 = (p_1 \\times p_2 \\times p_3 \\times \\dots \\times p_k) + 1$$\n\n*Awarded 2 marks (M1 for defining the product $N$, and M1 for constructing $M = N+1$).*"},{"type":"text","order":2,"title":"Analyze divisibility of M","content":"Let's consider what happens when we divide $M$ by any of the prime numbers $p_i$ in our set $P$:\n\n$$\\frac{M}{p_i} = \\frac{(p_1 \\times p_2 \\times \\dots \\times p_k) + 1}{p_i} = \\frac{p_1 \\times p_2 \\times \\dots \\times p_k}{p_i} + \\frac{1}{p_i}$$\n\nSince $p_i$ is a factor of the product, the first term is an integer, leaving a remainder of $1$ from the second term. \n\nTherefore, $M$ is **not divisible** by any of the primes $p_1, p_2, \\dots, p_k$.\n\n*Awarded 1 mark (A1) for showing that dividing $M$ by any prime in the finite set leaves a remainder of 1.*"},{"type":"text","order":3,"title":"Identify the contradiction","content":"By the Fundamental Theorem of Arithmetic, every integer greater than $1$ is either a prime number itself or can be uniquely factored into a product of prime numbers.\n\nSince $M > 1$:\n- If $M$ is a prime number, then we have found a prime number that is not in our original list $P$ (since $M$ is much larger than $p_k$).\n- If $M$ is not prime, it must have a prime factor. Since none of the primes in our list $P$ divide $M$, this prime factor must be some new prime number that is not in our list $P$.\n\nIn either case, there exists a prime number that is not included in our supposedly complete list $P$.\n\n*Awarded 1 mark (M1) for explaining that $M$ must either be prime or have a prime factor not in the original list.*"},{"type":"text","order":4,"title":"State the final conclusion","content":"This directly contradicts our initial assumption that $P = \\{p_1, p_2, \\dots, p_k\\}$ contains all prime numbers.\n\nSince our assumption leads to a contradiction, the assumption must be false. Therefore, Elena's claim is incorrect, and we conclude that **there are infinitely many prime numbers**.\n\n*Awarded 1 mark (A1) for concluding the proof by identifying the contradiction and stating the final result clearly.*"}]},{"id":"a0bd4ae7-e2d2-4ebf-8cfe-418bad7fef9f:factorial_construction","label":"Factorial-Based Construction","steps":[{"type":"text","order":0,"title":"State the initial assumption","content":"To prove Elena's claim is false using proof by contradiction, we must begin by assuming that her claim is true.\n\nAssume there is a finite set of prime numbers:\n\n$$P = \\{p_1, p_2, p_3, \\dots, p_k\\}$$\n\nwhere $p_k$ is the largest prime number in existence.\n\nThis earns the first mark **B1**.","highlights":[{"text":"Elena makes the claim that there is a finite set of prime numbers $P = \\{p_1, p_2, p_3, \\dots, p_k\\}$ where $p_k$ is the largest prime","location":"specification"}]},{"type":"text","order":1,"title":"Construct the integer $M$","content":"Using the **Factorial-Based Construction** approach, we define a new positive integer, $M$, by adding $1$ to the factorial of the largest prime, $p_k$:\n\n$$M = p_k! + 1$$\n\nwhere the factorial $p_k!$ is the product of all positive integers from $1$ up to $p_k$:\n\n$$p_k! = p_k \\times (p_k - 1) \\times (p_k - 2) \\times \\dots \\times 2 \\times 1$$\n\nThis step corresponds to the method marks **M1** and **M1**."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$M = \\htmlData{anchor=fact}{\\color{@sky}{p_k!}} + \\htmlData{anchor=one}{\\color{@amber}{1}}$"},{"text":"Since every prime $p_i$ in our finite set $P$ is less than or equal to $p_k$, it must be one of the integers in the product $p_k!$."},{"latex":"$$p_i \\text{ divides } \\color{@sky}{p_k!} \\implies \\frac{M}{p_i} = \\frac{\\color{@sky}{p_k!}}{p_i} + \\frac{\\color{@amber}{1}}{p_i}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"fact"},{"color":"amber","style":"text-color","anchor":"one"}],"connections":[{"to":"one","from":"fact","color":"sky","label":null,"style":"solid"}]},"order":2,"title":"Analyze divisibility of $M$","description":"Any prime $p_i$ in our assumed finite set must divide $p_k!$ because $p_i \\le p_k$. Dividing $M$ by $p_i$ will therefore always leave a remainder of $1$ (satisfying mark **A1**)."},{"type":"text","order":3,"title":"Examine the prime factors","content":"According to the Fundamental Theorem of Arithmetic, the integer $M$ must either be prime itself or it must be a product of prime factors.\n\n- If $M$ is prime, then since $M = p_k! + 1 > p_k$, we have found a prime number greater than $p_k$.\n- If $M$ is composite, it must have a prime factor, say $q$. Since $M$ is not divisible by any of the primes in our set $P$, this prime factor $q$ cannot be in $P$. Thus, $q$ must be greater than $p_k$ ($q > p_k$).\n\nIn either case, there exists a prime number greater than the assumed largest prime $p_k$ (earning mark **M1**)."},{"type":"text","order":4,"title":"Reach the contradiction","content":"Having found a prime number greater than $p_k$ directly contradicts our initial assumption that $p_k$ is the largest prime and that the set of prime numbers is finite.\n\nTherefore, our initial assumption must be false. Elena's claim is incorrect, and there are infinitely many prime numbers.\n\nThis final conclusion completes the proof and earns the final mark **A1**."}]}],"generatedAt":"2026-06-09T01:58:01.384Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":44},"walkthroughVideo":null},{"id":"fe9a1d93-a8a9-4e0d-aca3-70ae3b951191","revisionId":"7803b325-4a93-4e0d-bc4a-5f00793c42d8","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"hard","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"a6e52348-6ec1-465b-bf27-4ebc042ef84c","label":"a","content":"Prove by contradiction that $\\sqrt{3}$ is irrational.","markscheme":"- Assume $\\sqrt{3}$ is rational, so $\\sqrt{3} = \\frac{a}{b}$ where $a, b$ are integers with no common factors. B1\n- Square both sides and rearrange: $3 = \\frac{a^2}{b^2} \\Rightarrow 3b^2 = a^2$ M1\n- Deduce $a^2$ is a multiple of 3, which implies $a$ is a multiple of 3. A1\n- Let $a = 3k$ for some integer $k$, then $3b^2 = (3k)^2 = 9k^2 \\Rightarrow b^2 = 3k^2$ M1\n- Deduce $b^2$ is a multiple of 3, so $b$ is a multiple of 3. A1\n- Since $a$ and $b$ both have a common factor of 3, this contradicts the assumption that they have no common factors. Therefore $\\sqrt{3}$ is irrational. A1","marks":6,"order":0,"markingRubric":{"totalMarks":6,"points":[{"id":"p1","marks":1,"description":"Assume $\\sqrt{3} = a/b$ where $a, b$ are integers with no common factors.","code":"B"},{"id":"p2","marks":1,"description":"Square and rearrange to $3b^2 = a^2$.","code":"M"},{"id":"p3","marks":1,"description":"Correctly deduce that $a$ is a multiple of 3.","code":"A"},{"id":"p4","marks":1,"description":"Substitute $a = 3k$ and simplify to $b^2 = 3k^2$.","code":"M"},{"id":"p5","marks":1,"description":"Correctly deduce that $b$ is a multiple of 3.","code":"A"},{"id":"p6","marks":1,"description":"Conclude that the common factor contradicts the initial assumption.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-06T14:03:23.615Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"coprime_contradiction","label":"Coprime Fraction Method","steps":[{"type":"text","order":0,"title":"Assume the contrary","content":"To prove that $\\sqrt{3}$ is irrational by contradiction, we begin by assuming that it is actually rational.\n\nBy definition, any rational number can be expressed as a fraction in its simplest form:\n\n$$\\sqrt{3} = \\frac{a}{b}$$\n\nwhere $a$ and $b$ are integers with no common factors (meaning they are **coprime** and $b \\neq 0$).\n\n*(This set up earns the first **B1** mark)*"},{"type":"text","order":1,"title":"Square and rearrange","content":"To eliminate the square root, we square both sides of the equation:\n\n$$3 = \\frac{a^2}{b^2}$$\n\nNow, we multiply both sides by $b^2$ to rearrange the equation into a linear form:\n\n$$3b^2 = a^2$$\n\n*(This algebraic rearrangement earns the **M1** mark)*"},{"type":"text","order":2,"title":"Analyze the variable a","content":"Since $3b^2 = a^2$ and $b$ is an integer, it is clear that $a^2$ must be a multiple of $3$.\n\nBecause $3$ is a prime number, if the square of an integer is divisible by $3$, then the integer itself must also be divisible by $3$. Therefore, **$a$ is a multiple of $3$**.\n\n*(Deducing that $a$ is a multiple of $3$ earns the **A1** mark)*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$3b^2 = \\htmlData{anchor=target-a}{\\color{@sky}{a}}^2$"},{"latex":"$$\\text{Since } a \text{ is a multiple of } 3\text{, we let } \\htmlData{anchor=source-a}{\\color{@sky}{a}} = \\color{@amber}{3k} \\text{ for some integer } k$"},{"latex":"$$3b^2 = (\\color{@amber}{3k})^2 = 9k^2$"},{"latex":"$$b^2 = 3k^2$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"source-a"},{"color":"sky","style":"text-color","anchor":"target-a"}],"connections":[{"to":"target-a","from":"source-a","color":"sky","label":"substitute","style":"solid"}]},"order":3,"title":"Substitute to analyze b","description":"Since $a$ is a multiple of $3$, we can write it as $a = 3k$ where $k$ is an integer. Substituting this back into our original rearranged equation allows us to find the properties of $b$."},{"type":"text","order":4,"title":"Deduce b is a multiple","content":"From our simplified equation:\n\n$$b^2 = 3k^2$$\n\nWe can see that $b^2$ is a multiple of $3$.\n\nFollowing the same logic as before, since $3$ is a prime number and $b^2$ is divisible by $3$, it follows that **$b$ must also be a multiple of $3$**.\n\n*(Setting $a = 3k$ and simplifying earns the **M1** mark, and deducing $b$ is a multiple of $3$ earns the **A1** mark)*"},{"type":"text","order":5,"title":"Identify the contradiction","content":"We have shown that:\n- $a$ is a multiple of $3$\n- $b$ is a multiple of $3$\n\nThis means that both $a$ and $b$ share a common factor of $3$. However, this directly contradicts our initial assumption that $a$ and $b$ are coprime and share no common factors.\n\nSince our original assumption has led to a contradiction, it must be false. \n\nTherefore, **$\\sqrt{3}$ is irrational**.\n\n*(This concluding contradiction argument earns the final **A1** mark)*"}]},{"id":"prime_factorization","label":"Prime Factorization Method","steps":[{"type":"text","order":0,"title":"Set up the contradiction","content":"To prove that $\\sqrt{3}$ is irrational by contradiction, we begin by assuming the opposite: that $\\sqrt{3}$ is rational.\n\nIf $\\sqrt{3}$ is rational, it can be written as the ratio of two positive integers, $a$ and $b$:\n$$\\sqrt{3} = \\frac{a}{b} \\quad \\text{where } a, b \\in \\mathbb{Z}^+$$\n\nNow, square both sides to eliminate the square root:\n$$3 = \\frac{a^2}{b^2}$$\n\nMultiply both sides by $b^2$ to get a relation in terms of integers:\n$$3b^2 = a^2$$\n\n*This initial setup and rearrangement correspond to the **B1** and **M1** marks on typical exam markschemes.*","highlights":[{"text":"Prove by contradiction that $\\sqrt{3}$ is irrational.","location":"specification"}]},{"type":"text","order":1,"title":"Analyze the right side","content":"By the **Fundamental Theorem of Arithmetic**, every positive integer greater than 1 has a unique prime factorization. Let's analyze the multiplicity (the exponent) of the prime factor $3$ on both sides of our equation.\n\nLet the exponent of $3$ in the prime factorization of $a$ be $p$ (where $p \\ge 0$). \n\nWhen we square $a$, the exponents in its prime factorization are doubled. Thus, the exponent of the prime factor $3$ in the factorization of $a^2$ is:\n$$\\text{Exponent of } 3 \\text{ in } a^2 = 2p$$\n\nSince $2p$ is a multiple of $2$, the number of times $3$ appears in the prime factorization of $a^2$ must be **even**."},{"type":"text","order":2,"title":"Analyze the left side","content":"Now, let's look at the left-hand side of the equation, $3b^2$.\n\nLet the exponent of $3$ in the prime factorization of $b$ be $q$ (where $q \\ge 0$).\n\nWhen we square $b$, the exponent of $3$ in $b^2$ is doubled to $2q$ (an even number).\n\nWhen we multiply $b^2$ by $3$, we add exactly one more factor of $3$ to the prime factorization. Therefore, the total exponent of $3$ in the prime factorization of $3b^2$ is:\n$$\\text{Exponent of } 3 \\text{ in } 3b^2 = 2q + 1$$\n\nSince $2q + 1$ is of the form $2k + 1$, the number of times $3$ appears in the prime factorization of $3b^2$ must be **odd**."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$3b^2 = a^2$$"},{"latex":"$$$\\text{Exponent of } 3 \\text{ on LHS} = \\htmlData{anchor=lhs-exp}{\\color{@pink}{2q + 1}}$$"},{"latex":"$$$\\text{Exponent of } 3 \\text{ on RHS} = \\htmlData{anchor=rhs-exp}{\\color{@sky}{2p}}$$"}],"highlights":[{"color":"pink","style":"box","anchor":"lhs-exp"},{"color":"sky","style":"box","anchor":"rhs-exp"}],"connections":[{"to":"rhs-exp","from":"lhs-exp","color":"purple","label":"must be equal","style":"dashed"}]},"order":3,"title":"Find the contradiction","description":"For the equation $3b^2 = a^2$ to hold, the prime factorizations of both sides must be identical. This requires the multiplicity of the prime factor $3$ to be equal on both sides: \n\n$$2q + 1 = 2p$$\n\nHowever, $2q + 1$ is an **odd** integer and $2p$ is an **even** integer. An odd integer can never equal an even integer. \n\nThis contradiction proves that our initial assumption—that $\\sqrt{3}$ is rational—must be false. Therefore, $\\sqrt{3}$ is irrational."}]}],"generatedAt":"2026-06-09T11:50:33.540Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"a6e52348-6ec1-465b-bf27-4ebc042ef84c:coprime_contradiction","label":"Coprime Fraction Method","steps":[{"type":"text","order":0,"title":"Assume the contrary","content":"To prove that $\\sqrt{3}$ is irrational by contradiction, we begin by assuming that it is actually rational.\n\nBy definition, any rational number can be expressed as a fraction in its simplest form:\n\n$$\\sqrt{3} = \\frac{a}{b}$$\n\nwhere $a$ and $b$ are integers with no common factors (meaning they are **coprime** and $b \\neq 0$).\n\n*(This set up earns the first **B1** mark)*"},{"type":"text","order":1,"title":"Square and rearrange","content":"To eliminate the square root, we square both sides of the equation:\n\n$$3 = \\frac{a^2}{b^2}$$\n\nNow, we multiply both sides by $b^2$ to rearrange the equation into a linear form:\n\n$$3b^2 = a^2$$\n\n*(This algebraic rearrangement earns the **M1** mark)*"},{"type":"text","order":2,"title":"Analyze the variable a","content":"Since $3b^2 = a^2$ and $b$ is an integer, it is clear that $a^2$ must be a multiple of $3$.\n\nBecause $3$ is a prime number, if the square of an integer is divisible by $3$, then the integer itself must also be divisible by $3$. Therefore, **$a$ is a multiple of $3$**.\n\n*(Deducing that $a$ is a multiple of $3$ earns the **A1** mark)*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$3b^2 = \\htmlData{anchor=target-a}{\\color{@sky}{a}}^2$"},{"latex":"$$\\text{Since } a \\text{ is a multiple of } 3\\text{, we let } \\htmlData{anchor=source-a}{\\color{@sky}{a}} = \\color{@amber}{3k} \\text{ for some integer } k$"},{"latex":"$$3b^2 = (\\color{@amber}{3k})^2 = 9k^2$"},{"latex":"$$b^2 = 3k^2$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"source-a"},{"color":"sky","style":"text-color","anchor":"target-a"}],"connections":[{"to":"target-a","from":"source-a","color":"sky","label":"substitute","style":"solid"}]},"order":3,"title":"Substitute to analyze b","description":"Since $a$ is a multiple of $3$, we can write it as $a = 3k$ where $k$ is an integer. Substituting this back into our original rearranged equation allows us to find the properties of $b$."},{"type":"text","order":4,"title":"Deduce b is a multiple","content":"From our simplified equation:\n\n$$b^2 = 3k^2$$\n\nWe can see that $b^2$ is a multiple of $3$.\n\nFollowing the same logic as before, since $3$ is a prime number and $b^2$ is divisible by $3$, it follows that **$b$ must also be a multiple of $3$**.\n\n*(Setting $a = 3k$ and simplifying earns the **M1** mark, and deducing $b$ is a multiple of $3$ earns the **A1** mark)*"},{"type":"text","order":5,"title":"Identify the contradiction","content":"We have shown that:\n- $a$ is a multiple of $3$\n- $b$ is a multiple of $3$\n\nThis means that both $a$ and $b$ share a common factor of $3$. However, this directly contradicts our initial assumption that $a$ and $b$ are coprime and share no common factors.\n\nSince our original assumption has led to a contradiction, it must be false. \n\nTherefore, **$\\sqrt{3}$ is irrational**.\n\n*(This concluding contradiction argument earns the final **A1** mark)*"}]},{"id":"a6e52348-6ec1-465b-bf27-4ebc042ef84c:prime_factorization","label":"Prime Factorization Method","steps":[{"type":"text","order":0,"title":"Set up the contradiction","content":"To prove that $\\sqrt{3}$ is irrational by contradiction, we begin by assuming the opposite: that $\\sqrt{3}$ is rational.\n\nIf $\\sqrt{3}$ is rational, it can be written as the ratio of two positive integers, $a$ and $b$:\n$$\\sqrt{3} = \\frac{a}{b} \\quad \\text{where } a, b \\in \\mathbb{Z}^+$$\n\nNow, square both sides to eliminate the square root:\n$$3 = \\frac{a^2}{b^2}$$\n\nMultiply both sides by $b^2$ to get a relation in terms of integers:\n$$3b^2 = a^2$$\n\n*This initial setup and rearrangement correspond to the **B1** and **M1** marks on typical exam markschemes.*","highlights":[{"text":"Prove by contradiction that $\\sqrt{3}$ is irrational.","location":"specification"}]},{"type":"text","order":1,"title":"Analyze the right side","content":"By the **Fundamental Theorem of Arithmetic**, every positive integer greater than 1 has a unique prime factorization. Let's analyze the multiplicity (the exponent) of the prime factor $3$ on both sides of our equation.\n\nLet the exponent of $3$ in the prime factorization of $a$ be $p$ (where $p \\ge 0$). \n\nWhen we square $a$, the exponents in its prime factorization are doubled. Thus, the exponent of the prime factor $3$ in the factorization of $a^2$ is:\n$$\\text{Exponent of } 3 \\text{ in } a^2 = 2p$$\n\nSince $2p$ is a multiple of $2$, the number of times $3$ appears in the prime factorization of $a^2$ must be **even**."},{"type":"text","order":2,"title":"Analyze the left side","content":"Now, let's look at the left-hand side of the equation, $3b^2$.\n\nLet the exponent of $3$ in the prime factorization of $b$ be $q$ (where $q \\ge 0$).\n\nWhen we square $b$, the exponent of $3$ in $b^2$ is doubled to $2q$ (an even number).\n\nWhen we multiply $b^2$ by $3$, we add exactly one more factor of $3$ to the prime factorization. Therefore, the total exponent of $3$ in the prime factorization of $3b^2$ is:\n$$\\text{Exponent of } 3 \\text{ in } 3b^2 = 2q + 1$$\n\nSince $2q + 1$ is of the form $2k + 1$, the number of times $3$ appears in the prime factorization of $3b^2$ must be **odd**."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$3b^2 = a^2$$"},{"latex":"$$$\\text{Exponent of } 3 \\text{ on LHS} = \\htmlData{anchor=lhs-exp}{\\color{@pink}{2q + 1}}$$"},{"latex":"$$$\\text{Exponent of } 3 \\text{ on RHS} = \\htmlData{anchor=rhs-exp}{\\color{@sky}{2p}}$$"}],"highlights":[{"color":"pink","style":"box","anchor":"lhs-exp"},{"color":"sky","style":"box","anchor":"rhs-exp"}],"connections":[{"to":"rhs-exp","from":"lhs-exp","color":"purple","label":"must be equal","style":"dashed"}]},"order":3,"title":"Find the contradiction","description":"For the equation $3b^2 = a^2$ to hold, the prime factorizations of both sides must be identical. This requires the multiplicity of the prime factor $3$ to be equal on both sides: \n\n$$2q + 1 = 2p$$\n\nHowever, $2q + 1$ is an **odd** integer and $2p$ is an **even** integer. An odd integer can never equal an even integer. \n\nThis contradiction proves that our initial assumption—that $\\sqrt{3}$ is rational—must be false. Therefore, $\\sqrt{3}$ is irrational."}]}],"generatedAt":"2026-06-09T11:50:33.613Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":72},"walkthroughVideo":null},{"id":"6c4ee647-e514-46f6-88b7-ec0733bbe1ed","revisionId":"868fe90c-9d56-45ad-849c-8097c84eeb25","questionNumber":null,"questionType":"LA","specification":"","difficulty":3,"difficultyBand":"hard","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"06518a2f-569b-4542-8bc1-666bf1872fb0","label":"a","content":"Using the method of proof by contradiction, prove that there is no greatest prime number.","markscheme":"- Assume that there is a greatest prime number, $L$. B1\n- Consider the product of all prime numbers from 2 up to $L$: $X = 2 \\times 3 \\times 5 \\times \\dots \\times L$. M1\n- Define $Y = X + 1$. M1\n- Argue that $Y$ is not divisible by any prime $p$ such that $p \\le L$, as division by any such prime leaves a remainder of 1. A1\n- Deduce that $Y$ is either a prime number itself or it has a prime factor that is greater than $L$. M1\n- Conclude that this contradicts the assumption that $L$ is the greatest prime number, therefore no greatest prime exists. A1","marks":6,"order":0,"markingRubric":{"totalMarks":6,"points":[{"id":"p1","marks":1,"description":"Assume that a largest prime number $L$ exists.","code":"B"},{"id":"p2","marks":1,"description":"Form the product of all primes up to $L$.","code":"M"},{"id":"p3","marks":1,"description":"Form a new number by adding 1 to that product.","code":"M"},{"id":"p4","marks":1,"description":"Demonstrate that no prime up to $L$ is a factor of the new number.","code":"A"},{"id":"p5","marks":1,"description":"Establish that a prime exists that is not in the original list.","code":"M"},{"id":"p6","marks":1,"description":"Conclude that the existence of a greatest prime is impossible.","code":"A"}],"finalAnswers":["There is no greatest prime number"],"generatedAt":"2026-05-06T14:05:19.695Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"euclid_product","label":"Euclid's Product Construction","steps":[{"type":"text","order":0,"title":"Assume the opposite","content":"To prove this statement by contradiction, we begin by assuming the opposite of what we want to show.\n\nAssume that there is a finite number of primes, meaning there exists a **greatest prime number**, which we will call $L$.\n\nThus, our complete list of all prime numbers is:\n$$2, 3, 5, 7, \\dots, L$$"},{"type":"text","order":1,"title":"Construct the product","content":"Next, we construct a new integer, $Y$, by multiplying all of these prime numbers together and adding $1$.\n\nLet $X$ be the product of all prime numbers up to $L$:\n$$X = 2 \\times 3 \\times 5 \\times \\dots \\times L$$\n\nNow, we define $Y$ as:\n$$Y = X + 1 = (2 \\times 3 \\times 5 \\times \\dots \\times L) + 1$$"},{"type":"text","order":2,"title":"Test divisibility of Y","content":"Let's consider what happens when we divide $Y$ by any of the prime numbers, $p$, from our list ($2 \\le p \\le L$).\n\nSince $p$ is a factor of the product $X$, dividing $X$ by $p$ leaves no remainder. \n\nTherefore, dividing $Y = X + 1$ by $p$ will always leave a remainder of $1$:\n\n$$\\frac{Y}{p} = \\frac{X}{p} + \\frac{1}{p}$$\n\nThis means that $Y$ is **not divisible** by any prime $p \\le L$."},{"type":"text","order":3,"title":"Analyze the prime factors","content":"According to the Fundamental Theorem of Arithmetic, every integer greater than $1$ must either be a prime number itself or be divisible by a prime number. Let's analyze these two cases for $Y$:\n\n- **Case 1: $Y$ is prime**\n Since $Y > L$, we have found a prime number greater than $L$.\n\n- **Case 2: $Y$ is composite**\n Since $Y$ is not divisible by any prime up to $L$, any prime factor of $Y$ must be **greater than $L$**."},{"type":"text","order":4,"title":"Reach a contradiction","content":"In both cases, we have shown the existence of a prime number that is greater than $L$.\n\nThis directly contradicts our initial assumption that $L$ is the greatest prime number.\n\nTherefore, our initial assumption must be false. We conclude that **there is no greatest prime number** (meaning the number of primes is infinite)."}]},{"id":"factorial_construction","label":"Factorial Construction","steps":[{"type":"text","order":0,"title":"Assume a greatest prime","content":"To prove there is no greatest prime number by contradiction, we begin by assuming the opposite of what we want to prove.\n\nAssume that there **is** a greatest prime number, which we will call $L$.\n\nThis means that the set of all prime numbers is finite, and any prime number $p$ must satisfy $p \\le L$.\n\n*(This earns the first mark **B1**)*"},{"type":"text","order":1,"title":"Construct the factorial expression","content":"Consider the factorial of $L$, which is the product of all integers from 1 up to $L$:\n\n$$L! = L \\times (L-1) \\times (L-2) \\times \\dots \\times 2 \\times 1$$\n\nNow, let us define a new integer $N$ as:\n\n$$N = L! + 1$$\n\n*(Constructing this expression earns the **M1** marks for creating a product of terms plus one)*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=ndef}{N} = \\htmlData{anchor=lfact}{L!} + 1$"},{"text":"Let $p$ be any prime number such that $p \\le L$. Since $p$ is an integer in the range $1$ to $L$, it is a factor of $L!$."},{"latex":"$$\\frac{\\htmlData{anchor=ndiv}{N}}{\\htmlData{anchor=pdiv1}{\\color{@sky}{p}}} = \\frac{\\htmlData{anchor=ldiv}{L!}}{\\htmlData{anchor=pdiv2}{\\color{@sky}{p}}} + \\frac{1}{\\color{@sky}{p}}$"},{"text":"Since $p$ divides $L!$, the term $\\frac{L!}{p}$ is an integer, leaving a remainder of $1$ when $N$ is divided by $p$."}],"highlights":[{"color":"sky","style":"text-color","anchor":"pdiv1"},{"color":"sky","style":"text-color","anchor":"pdiv2"}],"connections":[{"to":"ndiv","from":"ndef","color":"purple","label":"substitute","style":"solid"},{"to":"ldiv","from":"lfact","color":"amber","label":null,"style":"solid"}]},"order":2,"title":"Analyze divisibility by primes","description":"We divide the entire expression for $N$ by a prime $p \\le L$ to inspect its remainder. Since the remainder is 1, $N$ is not divisible by any prime $p \\le L$. (This earns the **A1** mark)"},{"type":"text","order":3,"title":"Deduce a larger prime","content":"According to the Fundamental Theorem of Arithmetic, every integer greater than 1 is either a prime number itself or can be factored into prime numbers.\n\nSince $N > 1$, it must have at least one prime factor, let us call it $q$.\n\nFrom our previous step, we established that **no** prime $p$ less than or equal to $L$ can divide $N$. Therefore, the prime factor $q$ cannot be any of the primes less than or equal to $L$.\n\nThus, we must have:\n\n$$q > L$$\n\n*(This deduction earns **M1**)*"},{"type":"text","order":4,"title":"State the contradiction","content":"We have shown that $N$ must have a prime factor $q$ such that $q > L$.\n\nThis directly **contradicts** our initial assumption that $L$ is the greatest prime number.\n\nTherefore, our assumption must be false. We conclude that there is no greatest prime number.\n\n*(This final conclusion earns **A1**)*"}]}],"generatedAt":"2026-06-09T06:39:31.170Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"06518a2f-569b-4542-8bc1-666bf1872fb0:euclid_product","label":"Euclid's Product Construction","steps":[{"type":"text","order":0,"title":"Assume the opposite","content":"To prove this statement by contradiction, we begin by assuming the opposite of what we want to show.\n\nAssume that there is a finite number of primes, meaning there exists a **greatest prime number**, which we will call $L$.\n\nThus, our complete list of all prime numbers is:\n$$2, 3, 5, 7, \\dots, L$$"},{"type":"text","order":1,"title":"Construct the product","content":"Next, we construct a new integer, $Y$, by multiplying all of these prime numbers together and adding $1$.\n\nLet $X$ be the product of all prime numbers up to $L$:\n$$X = 2 \\times 3 \\times 5 \\times \\dots \\times L$$\n\nNow, we define $Y$ as:\n$$Y = X + 1 = (2 \\times 3 \\times 5 \\times \\dots \\times L) + 1$$"},{"type":"text","order":2,"title":"Test divisibility of Y","content":"Let's consider what happens when we divide $Y$ by any of the prime numbers, $p$, from our list ($2 \\le p \\le L$).\n\nSince $p$ is a factor of the product $X$, dividing $X$ by $p$ leaves no remainder. \n\nTherefore, dividing $Y = X + 1$ by $p$ will always leave a remainder of $1$:\n\n$$\\frac{Y}{p} = \\frac{X}{p} + \\frac{1}{p}$$\n\nThis means that $Y$ is **not divisible** by any prime $p \\le L$."},{"type":"text","order":3,"title":"Analyze the prime factors","content":"According to the Fundamental Theorem of Arithmetic, every integer greater than $1$ must either be a prime number itself or be divisible by a prime number. Let's analyze these two cases for $Y$:\n\n- **Case 1: $Y$ is prime**\n Since $Y > L$, we have found a prime number greater than $L$.\n\n- **Case 2: $Y$ is composite**\n Since $Y$ is not divisible by any prime up to $L$, any prime factor of $Y$ must be **greater than $L$**."},{"type":"text","order":4,"title":"Reach a contradiction","content":"In both cases, we have shown the existence of a prime number that is greater than $L$.\n\nThis directly contradicts our initial assumption that $L$ is the greatest prime number.\n\nTherefore, our initial assumption must be false. We conclude that **there is no greatest prime number** (meaning the number of primes is infinite)."}]},{"id":"06518a2f-569b-4542-8bc1-666bf1872fb0:factorial_construction","label":"Factorial Construction","steps":[{"type":"text","order":0,"title":"Assume a greatest prime","content":"To prove there is no greatest prime number by contradiction, we begin by assuming the opposite of what we want to prove.\n\nAssume that there **is** a greatest prime number, which we will call $L$.\n\nThis means that the set of all prime numbers is finite, and any prime number $p$ must satisfy $p \\le L$.\n\n*(This earns the first mark **B1**)*"},{"type":"text","order":1,"title":"Construct the factorial expression","content":"Consider the factorial of $L$, which is the product of all integers from 1 up to $L$:\n\n$$L! = L \\times (L-1) \\times (L-2) \\times \\dots \\times 2 \\times 1$$\n\nNow, let us define a new integer $N$ as:\n\n$$N = L! + 1$$\n\n*(Constructing this expression earns the **M1** marks for creating a product of terms plus one)*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=ndef}{N} = \\htmlData{anchor=lfact}{L!} + 1$"},{"text":"Let $p$ be any prime number such that $p \\le L$. Since $p$ is an integer in the range $1$ to $L$, it is a factor of $L!$."},{"latex":"$$\\frac{\\htmlData{anchor=ndiv}{N}}{\\htmlData{anchor=pdiv1}{\\color{@sky}{p}}} = \\frac{\\htmlData{anchor=ldiv}{L!}}{\\htmlData{anchor=pdiv2}{\\color{@sky}{p}}} + \\frac{1}{\\color{@sky}{p}}$"},{"text":"Since $p$ divides $L!$, the term $\\frac{L!}{p}$ is an integer, leaving a remainder of $1$ when $N$ is divided by $p$."}],"highlights":[{"color":"sky","style":"text-color","anchor":"pdiv1"},{"color":"sky","style":"text-color","anchor":"pdiv2"}],"connections":[{"to":"ndiv","from":"ndef","color":"purple","label":"substitute","style":"solid"},{"to":"ldiv","from":"lfact","color":"amber","label":null,"style":"solid"}]},"order":2,"title":"Analyze divisibility by primes","description":"We divide the entire expression for $N$ by a prime $p \\le L$ to inspect its remainder. Since the remainder is 1, $N$ is not divisible by any prime $p \\le L$. (This earns the **A1** mark)"},{"type":"text","order":3,"title":"Deduce a larger prime","content":"According to the Fundamental Theorem of Arithmetic, every integer greater than 1 is either a prime number itself or can be factored into prime numbers.\n\nSince $N > 1$, it must have at least one prime factor, let us call it $q$.\n\nFrom our previous step, we established that **no** prime $p$ less than or equal to $L$ can divide $N$. Therefore, the prime factor $q$ cannot be any of the primes less than or equal to $L$.\n\nThus, we must have:\n\n$$q > L$$\n\n*(This deduction earns **M1**)*"},{"type":"text","order":4,"title":"State the contradiction","content":"We have shown that $N$ must have a prime factor $q$ such that $q > L$.\n\nThis directly **contradicts** our initial assumption that $L$ is the greatest prime number.\n\nTherefore, our assumption must be false. We conclude that there is no greatest prime number.\n\n*(This final conclusion earns **A1**)*"}]}],"generatedAt":"2026-06-09T06:39:41.100Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":48},"walkthroughVideo":null}],"subjectCode":"a_level_maths__ocr","summaries":[{"id":"fd7ef5e5-e672-484d-827d-216a0bf187eb","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"004f1715-7f20-44bd-b303-beb33cb8f946","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"0f28af03-b719-4ce4-8b8a-9e1d7a79b4a7","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"135c4fa2-edaf-426c-aff5-6dff03228d23","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"26f0bc1d-bd36-470c-847c-ee92ef94d932","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"41bbfce2-2bcf-4f6a-b290-e3abcbb7a077","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"510dbdcb-d107-4e6d-ab01-cf07807fc6f9","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"6587ad34-dbe7-4eaf-93db-d0d216186419","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"6b925623-c577-4449-a2dd-7a78f36a658f","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"be93578c-b7d2-4df6-acd7-6ef98a572b2c","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"e56fe79b-a1e4-4b6d-9dca-a4042e1cf70b","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f5006b00-71ec-4fe1-bfd6-a3689efa8c22","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"fe9a1d93-a8a9-4e0d-aca3-70ae3b951191","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"b5b2f95d-72a6-49b2-9d75-4662197e54af","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"23ef56dd-b092-41bc-abe8-4a2e89a6592d","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"47c36499-dd4e-4e1f-92ce-655791bcc3c1","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"6c36ee4d-6b4b-4ff9-accd-9ae7dbccb7ec","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"6c4ee647-e514-46f6-88b7-ec0733bbe1ed","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"8dc028ae-d62c-450b-ace3-35c6f6f6a7f8","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"93607e6d-2a9b-443f-a9bb-40c9577b10e1","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"9e651ba3-893d-4d7b-a060-3cd50b14dc4b","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"a3db02dd-cb03-45ff-bdb6-a8d761189719","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"aa3791df-7481-43a1-b0b9-cb8e3bcc9c44","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"c7d9a82a-fa0f-4093-aeb3-1480c2a1ac5f","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"d4188522-4341-49ba-a1c1-b0d2cd27b056","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"fe45c078-ebe7-4562-88d0-081c3c872bf2","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"9b74fb76-e5d9-40d0-9ed3-b77c0f17ab33","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"13c9487f-fdfe-413a-bd71-4821d6b1f4c3","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"2228aba7-e076-4386-af3a-881069ea3ae9","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"3f64531b-355c-4f81-9254-e606a290abcd","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"4470672a-f3bb-48d3-96e8-994a0a98913b","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"5198bcf0-d6eb-42d9-8112-78ad2e6eaba6","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"6c7892f5-02da-412a-8c42-cc99863a9e88","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"8ca4ab45-dbd5-45b2-958e-815a8065d7c6","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"9121389d-68f4-4fe8-b1ac-5609535df620","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"99118445-8ab0-44a7-80de-6bf301bd4ed0","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"cd862080-eb92-428a-a568-4df58ad39380","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"cef11906-7830-41ee-a754-d85efa8f052e","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"fe7a6f3f-171d-45bc-92d9-1de3f6bfdcee","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"9c62f3fe-64d0-47f3-b702-a3a3e8ff75ae","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"16d05447-9328-4f4f-8b04-cfecf208a89a","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"d071f313-7932-4ad0-aab3-9f797eda3853","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"6bf375b8-ee1c-473b-81a2-7e09b5dab084","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e422d31a-b12d-4fc1-944e-8dde69e2113c","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"89d385f6-788d-4b29-8eb9-98e58f6cdf0b","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"16ae10ed-591f-4bc2-9325-d69489e15934","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"163a5d57-77a2-4ba5-953f-1b38f65f8c9e","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"242514c0-179f-43d9-8f6f-6909d182f51f","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"04be3b2f-ee91-4e8e-aee3-c7538271be15","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"0facf6a1-ce05-494a-a17f-d1e24cc5518d","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"110f33b5-c52d-4784-96fb-188f2dbcc3d7","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"18ada11b-472e-44dd-8867-c451904a517a","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"221a8bcc-2438-4aa5-9af2-a39c38971592","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"24e423bc-3e1f-4e0f-a5c0-560cbaf277d9","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"28ff3fa0-a34c-47b1-a0da-e821e3dc1d38","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"2a409465-939f-416b-9cde-8ab37e855f68","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"2f78ea22-8340-4230-ad26-c371057132cf","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"30867d72-5744-4bfc-8cb8-ad6997d311ea","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"32d377a6-7e21-4be7-b952-8d7abc33cb85","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"335f9cac-0fa2-4509-b809-57630ededfff","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"380d8433-3d23-41bc-9b56-96df9a7d3e2b","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"3989536d-520e-433b-a640-28d6948a0c5d","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"3e0420d1-3854-4158-a1ca-3db7c0bb5383","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"3f5dfa65-d35a-4dc6-b83c-24ec3a347c5d","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":true},{"id":"3fc583bb-f920-4ef7-9d09-f8c7cf7a28c8","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"42297193-1333-43f2-aa61-ae3fa3307b39","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"4a91330d-930c-4f9b-ad01-662049f645a1","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"4b909e4c-24ab-43c5-9c0c-4120c12a3030","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"556163c1-5661-443f-9a3a-18d926fa39a0","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"55ce197c-d30d-4299-b46e-0fcccbaf663a","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"5980d418-006d-4ce1-9f63-7b5328010c01","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"59fd7d5a-d4d9-461f-b440-b94339dd8d64","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"627d78fa-3b88-4bf9-b8c7-bce4d034d9ef","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"671e773e-16b4-4a30-a467-e17fc654b8f1","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"6783b9fd-dd5e-43af-80ba-899510b6566e","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"6a6a3ade-c77a-4fc4-bc26-74bf76e58e6d","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"7175b86e-9fff-43b4-a327-41703bba4199","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"7507acb4-cc2e-4a16-9d97-2dd4b8c3c7b9","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"7824cce4-cdc6-48e1-b186-a9dcdc172e79","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"78ece143-0564-451e-bf6c-2923f38164f3","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"7ab76ecd-976a-485e-b4fa-57191d6ef0b3","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"7b4d24e7-ab8c-46d9-abf2-a96ab239f484","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"7c3def0f-80a7-43dc-a681-5029075dc4ac","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"7d969d3a-9fc7-4d26-ab86-c45c0b8ba0de","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"848766c4-ab44-48b4-8b99-de0761912209","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"8abefdd6-fbf2-4b4d-a23a-85262f20fe27","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"8e8ef60f-c37c-474f-b8ca-4c68daeeb75f","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"999aa15f-87c9-4201-9c01-48914ab348df","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"9d70be20-7d2f-4ba4-abe4-81fe965b006e","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"9fe290e1-2433-4a33-b885-81ce7d4d4972","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"a30286d7-5984-4bd6-aa68-ad8db5cb5584","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"adaa56f4-551d-4bdc-9b73-45f3148bd551","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"b67c9afd-5e2b-441d-8be2-91adbf81317c","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"b7f385f4-7aab-4565-8fdb-c7e461c7e4f3","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"b7fbf628-3084-47f8-8f14-a603e293136e","difficultyBand":"hard","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"b9facce6-e77c-47a1-ae79-3c336b9eb192","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"ca749a66-7845-4abd-9efb-f717cd52b067","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"d5a1ffe9-fb01-4354-8faa-227ea2a944f7","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"d6b1d512-3484-4b9a-b965-7f69406f2fe0","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"d9c2f77c-1079-4219-90d7-82ec2213012e","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"e2a673a1-9266-4837-90f6-f29a690b2107","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"ef88d367-8d71-41e2-9443-fc32c8140de7","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"f1574cc4-4197-4312-8406-33fa51e7701c","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f6c128b2-2f8d-4f0b-ad82-f672fd179ac4","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"f77fc775-be8f-46a1-a32d-54e49082f630","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"f8ecec2a-ee26-4f16-9936-e82039f53785","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"fd93ba39-853b-4ec4-9b28-6f485059e32c","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true}],"title":"1.1.4 Proof by contradiction (A-level only)","topicId":"b00a295e-cd21-4677-87b5-c55f0941d173","topicSidebar":"$L3c","topicSlug":"proof-by-contradiction","topicTitle":"1.1.4 Proof by contradiction (A-level only)"}]}],"$L3d","$L3e"]}]

1.1.4 Proof by contradiction (A-level only)

Question 4

1.1.4 Proof by contradiction (A-level only) Questions

1.1.4 Proof by contradiction (A-level only)

Question 4

1.1.4 Proof by contradiction (A-level only) Questions