Since the gcd$(a,b) = p$, where $p$ is prime, calculate the $\gcd(a^3,b)$ and the $\gcd(a^2,b^3)$

Question

My attempt

$p\mid a$ and $p\mid b,$ then

$a = pq_1$ , $b = pq_2$

$\gcd(a^2,b)$

$a^2 = (p^2 * q_1^2)$ ,$\ b = p*q_2$

To find the $\gcd(a^2,b)$, I have to do the euclidean algorithm:

$(p^2q_1^2)\bmod (pq_2)$

From there I can't Go on...

Answer 1

You know that $p$ divides both $a$ and $b$, but no other prime does, and $p^2$ doesn't divide both. So $\gcd(a^3, b)$ could be either $p$, $p^2$ or $p^3$, and $\gcd(a^2,b^3)$ could be either $p^2$ or $p^3$.