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I think I have an idea for this, but I'm really struggling on how to put this proof into words. Its clearly the $\text{lcm}(n,m)$. I'm just struggling to communicate why.

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You have to show that if $a$ is in the group then ${\rm lcm}(m,n)a=0$, and you have to find an element $a$ such that $ra\ne0$ for $0\lt r\lt{\rm\ lcm}(m,n)$. Can you work out how to write those out?

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Hint $\rm\ (0,\color{#c00}0) = k\cdot(1,\color{#c00}1) = (k,\color{#c00}k)\iff m\mid k,\, \color{#c00}{n\mid k}\iff lcm(m,n)\mid k,\:$ by definition of lcm.

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