Prove that the square of any positive integer of the form 5m+1 will leave a remainder 1 when divided by 5 for some integer m

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Solution

Given a positive integer of the form $5 m + 1$
Say
$N = 5 m + 1 N^{2} = 25 m^{2} + 1 + 10 m = 25 m^{2} + 10 m + 1 = 5 (5 m^{2} + 2 m) + 1 = 5 k + 1$
where $k = 5 m^{2} + 2 m =$ same integer
$N^{2} = 5 k + 1$
From the above we can infer that $N^{2}$ , of the form $5 k + 1$ , leaves a remainder of $^{'} 1^{'}$ when divided by $5$ .

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