I'm trying to show the statement in the title (where $\mathcal{P}$ is the set of prime numbers, and $N$ is viewed as a $\mathbb{Z}$-module). The idea is that every element of $N$ has only finitely many nonzero entries, so their order is at most their least common multiple. However, take the element $n = (0 \mod 2, 2 \mod 3, 0\mod 5, 4 \mod 7, 0 \mod p)_{p \in \mathcal{P}}$. Note that lcm(2,4) = 8.
Then
$8n = 8(0 \mod 2, 16 \mod 3, 0\mod 5, 32 \mod 7, 0 \mod p)_{p \in \mathcal{P}} = (0 \mod 2, 1 \mod 3, 0\mod 5, 4 \mod 7, 0 \mod p)_{p \in \mathcal{P}} \neq 0_N$.
I had expected to get $0_N$, so where is my reasoning faulty?
The order of a finite family of elements with finite order is the l.c.m. of the orders of each, not the l.c.m. of the elements.