Exercise 1.2.22 (Square of any integer of the form $5k+1$)

Question

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

Richard Ganaye · Accepted Answer

\begin{proof} If $n = 5k + 1$ for some integer $k$, then
\begin{align*}
n^2 &= 25 k^2 + 10k + 1\
&=5(5k^2 + 2k) + 1\
&= 5K + 1,
\end{align*}
where $K = 5k^2 + 2k$ is an integer, so $n^2$ is of the same form.
\end{proof}

Exercise 1.2.22 (Square of any integer of the form $5k+1$)

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Exercise 1.2.22 (Square of any integer of the form $5k+1$)

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