Suppose that $I$ and $J$ are ideals of $\mathbb{Z}$, with $I=(m)$ and $J=(n)$. This question has two parts:
1) Let r be the least common multiple of $m$ and $n$. Show that $I\cap J = (r)$.
2) Let $d=(m,n)$. Show that $I+J = (d)$.
Suppose that $I$ and $J$ are ideals of $\mathbb{Z}$, with $I=(m)$ and $J=(n)$. This question has two parts:
1) Let r be the least common multiple of $m$ and $n$. Show that $I\cap J = (r)$.
2) Let $d=(m,n)$. Show that $I+J = (d)$.