Consider two non-trivial subgroups $H_1,H_2$ of $\mathbb Z$. $H_1=m\mathbb Z, H_2=n\mathbb Z$ . We know that the intersection $H_1 \bigcap H_2$ is a subgroup, and therefore can be written as $t\mathbb Z$ for some non-negative integer $t$. Prove that $t = lcm(m,n)$. This proves that the intersection of two non-trivial subgroups of $\mathbb Z$ is again non-trivial.
This is what I tried so far: I assumed that the Lagrange theorem applied for the number of subgroups as well. Since the intersection of the two subgroups must include both of the subgroups, the number of subgroups $m,n$ must divide $t$. In order to fulfill this condition, $t$ must be $lcm(m,n)$. I’m just in doubt because I don’t think this is how I’m supposed to solve this question.