1. Let $p$ be a prime, $\left(a, p^{2}\right)=p,\left(b, p^{3}\right)=p^{2}$. Find $\left(a b, p^{4}\right), \quad\left(a+b, p^{4}\right)$.
1. $\begin{aligned} a=p a_{1}, b=p^{2} b_{1},\left(a_{1}, p\right) & =\left(b_{1}, p\right)=1 .\left(a b, p^{4}\right)=p^{3}\left(a_{1} \cdot b_{1}, p\right)=p^{3} . \\ \left(a+b, p^{4}\right) & =p\left(a_{1}+p b_{1}, p^{3}\right)=p .\end{aligned}$