For $|a|=24$ and $|b|=10$, find the possibilities for $|⟨a⟩∩⟨b⟩|$

Question

Let $a$ and $b$ belong to a group. If $|a|=24$ and $|b|=10$, find the possibilities for $|⟨a⟩∩⟨b⟩|$

I know that if $H$ and $K$ are subgroups of $G$,then $H∩K$ is a subgroup of $G$.

But I'm stuck as to how to use this here to find the answer.

Answer 1

Hint: Since $\langle a \rangle \cap \langle b \rangle$ is a subgroup of both $\langle a \rangle$ and $\langle b \rangle$, then $|\langle a \rangle \cap \langle b \rangle| \mid |a|$ and $|\langle a \rangle \cap \langle b \rangle| \mid |b|$. Therefore, by definition $|\langle a \rangle \cap \langle b \rangle| \mid (|a|,|b|)$ where $(|a|,|b|)$ denotes the greatest common divisor of $|a|$ and $|b|$. Can you proceed from here?