How to solve $ab^2+b+7\mid a^2b+a+b$, $(a,b)=?$

Question

$ab^2+b+7\mid a^2b+a+b$, $(a,b)=?$

Would you give me a hint how to solve this problem?

Answer 1

Hint: use the elementary property of divisibility relation $x,y,z,a,m\in \mathbb{Z}$:

\begin{eqnarray*} x\mid x \;\;\;\;\;\forall x\in \mathbb{N}\\ x\mid y \;\;\wedge y\mid x \implies x=y\\ x\mid y \;\;\wedge y\mid z \implies x\mid z\\ x\mid y \implies x\mid my\\ a\mid x \;\;\wedge a\mid y \implies a\mid x\pm y\\ \end{eqnarray*}

However, you can find a solution to this problem here: https://artofproblemsolving.com/community/c6h18491p124428

Answer 2

Hint: if $ab^2+b+7$ divides $a^2b+a+b$, then it also divides $b(a^2b+a+b)-a(ab^2+b+7)$.