Are Abelian Groups of Relatively Prime Orders Isomorphic?

[SMA_01]

Homework Statement

Let m and n be relatively prime positive integers. Show that if there are, up to isomorphism, r abelian groups of order m and s of order n, then there are rs abelian groups of order mn.

Homework Equations

The Attempt at a Solution

I'm not sure how to go about this. I was thinking of saying that since m and n are relatively prime, the gcd(m,n)=1; wouldn't this then imply that the group order would be mn? Because mn is the lcm of m and n?

Any help is appreciated.

 

[Karamata]

$$\hbox{gcd}(m,n)=1 \Rightarrow C_{mn}\cong C_{m} \times C_{n}$$.

HINT:1. If $$C_m=<a>, C_n=<b>$$ then prove that $$C_{m} \times C_{n}=<d>$$ where $$d=(a,b)$$
2. Cyclic groups same orders are isomorphic.