Show that if 7 | x^2 + 1, then 13 | x^3 + 5x^2 +17x - 100.
I'm really not sure how to do this other than use the definition of divisibility, though I don't know where to go after that.
If we had $\displaystyle x^2\equiv{-1}(\bmod.7)$ (1) we see immediately that $\displaystyle (x,7)=1$ and then, by Fermat's little Theorem (*): $\displaystyle x^6\equiv{1}(\bmod.7)$, however, (1) implies: $\displaystyle x^6=(x^2)^3\equiv{(-1)^3}={-1}(\bmod.7)$ which is a contradiction!
In fact there exists $\displaystyle x\in \mathbb{Z}$ such that $\displaystyle p|(x^2+1)$ ( p is a prime with $\displaystyle p>2$) if and only if $\displaystyle p\equiv{1}(\bmod.4)$