If $$m$$ is a positive integer then the values of $$k$$ for which $$6m + k$$ cannot be a perfect square are

Why perfect squares mod 6 ∈ {0, 1, 3, 4}: every integer leaves a remainder $$r \in \{0, 1, 2, 3, 4, 5\}$$ when divided by 6. Squaring each:

The distinct residues are $$\{0, 1, 3, 4\}$$. The residues 2 and 5 are never squares mod 6.

Since $$6m + k \equiv k \pmod 6$$, the expression cannot be a perfect square iff $$k \bmod 6 \in \{2, 5\}$$.

For $$k \in \{1, 2, 3, 4, 5\}$$ (the values present in the options): the impossible ones are 2 and 5.