I have two questions:

* Suppose $a,b,s,t,u,v ∈ \mathbb{Z}$ such that $sa + tb = 21$ and $ua + vb = 10$. Show that $gcd(a,b) = 1.$

I feel like I'm going about this one in the wrong way. We haven't covered this material in class yet, all I know is that when $gcd(a,b) = 1$, it means that $a$ and $b$ are coprime. I tried substituting parts of each equation into each other but I don't really get any meaningful progress that way.

* given $p ∈ PRIME$, $gcd(a, p^2) = p$ and $gcd(b, p^3) = p^2$, find:

a. $gcd(a+b, p^4)$

b. $gcd(ab, p^4)$