I'm trying to find the tensor product $\mathbb{Z}/m\mathbb{Z}\otimes \mathbb{Z}/n\mathbb{Z}$ using this theorem. Now I know what the end product is (namely: $\mathbb{Z}/\gcd(m,n)\mathbb{Z}$) and I proved it differently. However,
how do I get that $m\mathbb{Z}\cdot \mathbb{Z}/n\mathbb{Z}\cong \gcd(m,n)\mathbb{Z}/n\mathbb{Z}$?