That's supposed to be , not , correct?

You can't express the orders of the primes like that... at least, not assuming and are prime. For example, gives you a prime.

The orders of are respectively. So the direct product has order . Also, notice that, since are odd, each of is even.

We can actually prove a stronger result here:

If are two groups such that arenotrelatively prime, then the direct product cannot be cyclic.

Notice that this would settle this problem as well. Can you prove this? (Consider the effect of raising an element of to the power of .)