Can we characterize the pairs of positive integers $b<a$ such that: $b|(a^2+\gcd(a,b))$

Question

Can we characterize the pairs of positive integers $b<a$ such that:

$b|(a^2+\gcd(a,b))$

Answer 1

Let $d:=gcd(a,b)$, $a=da_0$ and $b=db_0$.

Then the condition given is $db_0|d^2a_0^2+d$. This is equivalent to $b_0|da_0^2+1$.