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algebraic expression for the sum of the squares of two odd integers.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies the expression and factors out 2 to show it is even.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-07T02:12:13.876Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"algebraic_parity_proof","label":"Algebraic parity proof","steps":[{"type":"text","order":0,"title":"Represent odd numbers","content":"Any odd positive integer can be written as $2k+1$, where $k$ is a non-negative integer. Therefore, let the two odd positive integers be $2n+1$ and $2m+1$, where $n$ and $m$ are non-negative integers. This sets up the proof; the required sum is written in the next step."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$S=\\htmlData{anchor=sum}{\\color{@sky}{(2n+1)^2+(2m+1)^2}}$"},{"latex":"$$S=\\htmlData{anchor=expanded}{\\color{@orange}{4n^2+4n+1+4m^2+4m+1}}$"},{"latex":"$$S=4n^2+4n+4m^2+4m+2$"},{"text":"The first line is the required expression for the sum of the squares, so it earns the M1 mark. Expanding and simplifying to the final line is the first part of the A1 work."}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"sum"},{"color":"orange","style":"text-color","anchor":"expanded"}]},"order":1,"title":"Expand the squares","description":"Write the sum of the squares, then expand each bracket and collect the constant terms."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$S=\\color{@sky}{\\htmlData{anchor=unfactored}{4n^2+4n+4m^2+4m+2}}$"},{"latex":"$$S=\\color{@green}{\\htmlData{anchor=factored}{2(2n^2+2n+2m^2+2m+1)}}$"},{"text":"The expansion shown in the previous step and the factorisation shown above together earn the A1 mark. 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