$x^{13}+x+90$ is divisible by $x^2-x+a$ $(a\in\mathbb N)$. Find $a$

Hint: Set $x=0$ then $a$ must divide $90$. Also setting $x=1$ shows that $a$ must divide $92$; so $a$ must divide the difference $2$.

The reason this works is because $x^2-x+a = x(x-1)+a$ so it is natural to find conditions on $a$ by setting $x=0,1$ so the first term drops out.

Set $x = 0$ hence $a \mid 90$

Set $x = 1$ hence $a \mid 92$

Find gcd of $90,92$. This is $2$, which means $a$ can be $1$ or $2$

Finally (credit to @fleablood) set $x = -1$. Note that $a+2$ has to divide $88$, so $a = 2$.