Prove that the square of any given odd number gives a remainder of 1 when divided by 4 – math.stackexchange.com

Prove that the square of any given odd number gives a remainder of 1 when divided by 4 – math.stackexchange.com

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I'm supposed to prove this by induction, and I get that if I assume $n$ is odd such that $n^2$ gives a remainder of 1, then $n^2=4m+1$ for a number $m$ where $m \in \mathbb{Z}$, but I honestly can't ...

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