A student was asked to prove, for $n\in {\mathbb{Z}}^{+}$, that

“if $n^2$ is a multiple of 5, then $n$ must be a multiple of 5”

The start of the student's proof by contradiction is shown in the box below.

Assumption: There exists an integer $n\in {\mathbb{Z}}^{+}$, such that $n^2$ is a multiple of 5, and $n$ is NOT a multiple of 5.

Case 1: Let $n=5k+1$ for some integer $k$.

$$n^2=(5k+1{)}^2=25k^2+10k+1=5(5k^2+2k)+1$$

which is not a multiple of 5.

Case 2: Let $n=5k+2$ for some integer $k$.

$$n^2=(5k+2{)}^2=25k^2+20k+4=5(5k^2+4k)+4$$

which is not a multiple of 5.