TITLE: A finite abelian group whose order is divisible by 10 contains an element of order 10
QUESTION [5 upvotes]: It is given that the order of some finite abelian group is divisible by 10. Prove that the group has a cyclic subgroup of order 10. 
It is clear that since order of group is divisible by 10. By converse to Lagrange's Theorem, 
if 10 divides the order of the group G, then G has a subgroup of order 10. 
But to ensure that this subgroup is cyclic.

REPLY [3 votes]: Hint:If $p$ and $q$ are distinct primes then $\mathbb{Z}_p\times\mathbb{Z}_q\cong \mathbb{Z}_{pq}$.