Verify each of the following equalities for subgroups of $\mathbb Z$: $\langle24, -36, 54\rangle=\langle6\rangle$

Question

Verify each of the following equalities for subgroups of $\mathbb Z$:
$$\langle24, -36, 54\rangle=\langle6\rangle$$

I'm trying to move ahead in my homework so I haven't worked this kind of problem yet. I read something about it having to be the lowest common multiple, which is why I'm assuming it is $\langle6\rangle$ but I need a better understanding of what to do here.

Answer 1

Since $24$, $-36$ and $54$ are all multiples of $6$, then $\langle 24,-36,54\rangle \subseteq \langle 6 \rangle$.

Conversely, $6 = 24 + (-36) + (-36) + 54$, so $6 \in \langle 24,-36,54\rangle$, from which it follows that $\langle 6 \rangle \subseteq \langle 24,-36,54\rangle$.

Answer 2

Hint: Prove both inclusions. One is easy. For the other, it suffices to prove that $6 \in \langle 24,−36,54 \rangle$.