Does there exist a natural number $a$ such that $a^2+1$ is divisble by $9$?

Question

Can the above question be solved? Or can it be proved that it can not be solved?

What is the best approach to solving such questions?

Answer 1

Take $a$ and suppose $9$ divides $a^2+1$. Then so does $3$. This means that $$a^2+1 \equiv 0 \mod{3}$$

and this cannot happen, since squares modulo $3$ are just $0$ and $1$.

So there are no integers $a$ such that $9$ divides $a^2+1$.