From an homework question that comes as an introduction to abelian groups. Regarding my efforts to solve the question, I have been trying to utilize the fundamental theorem of finite abelian groups, or that:

$G \cong \mathbb{Z}\_{k\_1} \bigoplus \mathbb{Z}\_{k\_2} \ldots \bigoplus \mathbb{Z}\_{k\_n}$

Wherein $k\_1 \ldots k\_n$ are powers of primes and $k\_i | k\_{i+1}$ for $0 < i \leq n$. It was the latter of those two properties that I felt was pertinent to the question. However, after playing around prime powers and factorizations, I did not conclude anything relevant to the original question. I sense I am not approaching this as I should be, because this property seems genuinely simple.

Is there some other manipulation of properties of abelian groups from which this falls out straightforwardly, or alternatively am I not properly understanding the fundamental theorem?