Finitely Generated Abelian Groups

I'll be glad to receive some help in the following question:

Let be prime and let be non-negative integers. Show that if :

then the integers are uniquely determined by G . (Hint: consider the kernel of the homomorphism that is multiplication by . Show that determine . Proceed similarly )

I've tried considering the mentioned homomorphisms, but without any success... I'll be delighted to receive some guidance/solution to this problem (that is some kind of a preliminary step towards the proof of the classification theorem of abelian groups).

Thanks in advance !