Exponent of a direct product of groups Prove that if the group $G=\prod_{i=1}^nH_i$, where each $H_i$ is a finite group, then the exponent of $G$ which is
$\exp(G)=\min\{n \in \mathbb{N}:g^n=e, \forall g \in G\}$ is equal with $\operatorname{lcm}(\exp(H_1),\ldots,\exp(H_n))=M$
I proved that $\exp(G) \leqslant M$
Can someone help me with the other direction ?
Thank you in advance!

You show the other direction by identifying an element of the product which has order M.  Hint: Find a tuple that maximizes each coordinate.
