<s>[INST] Why is $x^2+1$ divisible by $10$ if $x$ has a $3$ or $7$ in the one's place? So I have the simple polynomial $x^2+1$. If I plug in ANY number that has a $3$ or a $7$ in the ones place $x^2+1$ is divisible by $10$. Why? Is there a reason for this?
So numbers like $3,7,13,17,23,27,\ldots$ when plugged into $x^2+1$ is divisible by 10. Why? Is there a reason for this?
 [/INST] \begin{align*}
(10k + 3)^2 + 1 &= 100k^2 + 60 k + 9 + 1 = 10(10k^2 + 6k + 1)
\end{align*}
and similarly for $7$.
</s>