As the question says. I'm having trouble putting all the pieces together. I've shown that $m$ divides $n$ if and only if $n \Bbb Z$ is a subring of $m \Bbb Z$ and vice versa. does this suffice to show that $n \Bbb Z + m \Bbb Z = d \Bbb Z$ where d is some divisor of $m$ and $n$, and then show that $d$ must be the greatest common divisor? I'm struggling to see what exactly I need to prove/how to put them together.