Prove that the square of any positive integer is of form $5 q, 5 q + 1, 5 q + 4$ for some integer $q$ .

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Solution

Let x be any positive integer
then, $x = 5 q$ or $x = 5 q + 1$ or $x = 5 q + 4$ for integer $x$
If $x = 5 q$
$x^{2} = (5 q)^{2} = 25 q^{2} = 5 (5 q^{2}) = 5 n$
where $n = 5 q^{2}$
If $x = 5 q + 1$
$x^{2} = (5 q + 1)^{2} = 25 q^{2} + 10 q + 1 = 5 (5 q^{2} + 2 q) + 1 = 5 n + 1$
where $n = 5 q^{2} + 2 q$
If $x = 5 q + 4$
$x^{2} = (5 q + 4)^{2} = 25 q^{2} + 40 q + 16 = 5 (5 q^{2} + 8 q + 3) + 1 = 5 n + 1$
where $n = 5 q^{2} + 8 q + 3$
$∴$ In each three cases $x^{2}$ is either of the form $5 q$ or $5 q + 1$ or $5 q + 4$ and for integer q.

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