Question:

If p and q are odd primes and m and n are positive integers show why $\displaystyle U(p^m) \oplus U(q&n) $ is not cyclic

where U(n) = { a < n | gcd(a,n)=1}

Attempt:

I know that the group is isomorphic to $\displaystyle U(p^m*q^n)$ so if I can show that one is not cyclic I can get the answer.

It must somehow involve the fact that p and q are odd

so I said p=2k+1 and q = 2L+1 where k and and L are primes. Then I know the order of the group is (2k+1)^(m-1)*2k*(2k+1)^(n-1)*(2L)

but I do not know where to go from there. Any tips?