Prove the LCM of an element in a group

If $\displaystyle G_1, G_2, . . . , G_n$ are finite groups, prove that the order of $\displaystyle (a_1, a_2, . . . , a_n)$ in $\displaystyle G_1 \times G_2 \times ... \times G_n$ is the LCM of the orders $\displaystyle |a_1|, |a_2|, ..., |a_n|$.

Attempt at the solution:

Since $\displaystyle G_1, G_2, . . . , G_n$ are finite groups, we know by definition that $\displaystyle G_1 \times G_2 \times ... \times G_n$ has order $\displaystyle |G_1| \cdot |G_2| \cdot \cdot \cdot |G_n|$.

Let $\displaystyle (a_1, a_2, . . . , a_n) \in G_1 \times G_2 \times ... \times G_n$. Then $\displaystyle |(a_1, a_2, . . . , a_n)| = |a_1| \cdot |a_2| \cdot \cdot \cdot |a_n|$.

From this point I think it's obvious enough that it's the LCM, but I don't know if it's sufficient. Could anyone help me finish this to make it more formal if possible?