Prove that the square of any positive integer of the form $5 q + 1$ is of the same form.

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Solution

Let $n = 5 q + 1.$ Then,
$n^{2} = 25 q^{2} + 10 q + 1 = 5 (5 q^{2} + 2 q) + 1 = 5 m + 1,$ where $m = 5 q^{2} + 2 q$
$\Rightarrow n^{2}$ is of the form $5 m + 1.$

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