if r belongs integer and r is a non zerosolution of x^2+ax+b=0 (where a,b belongs to integers), prove that r|b.
We are told that $\displaystyle r^2 + ar + b = 0$
$\displaystyle \Rightarrow r^2 + ar = -b$
$\displaystyle \Rightarrow r(r + a) = -b$
Now what can you say?
Spoiler:
Since $\displaystyle r,a \in \mathbb{Z}$, we have that $\displaystyle (r + a) \in \mathbb{Z}$, so that $\displaystyle r$ divides the left side of the equation. But this means it must also divide the right side. Hence, $\displaystyle r|b$
Since $\displaystyle r,a \in \mathbb{Z}$, we have that $\displaystyle (r + a) \in \mathbb{Z}$, so that $\displaystyle r$ divides the left side of the equation. But this means it must also divide the right side. Hence, $\displaystyle r|b$