This implication isn't correct. It's true that if a prime divides a product of two numbers, then it must divide at least one of them. The same isn't true, however, for prime powers. For example, , but 9 doesn't divide either factor.which implies that either or .
This point is the crux of the problem. Please think about it. Write out some examples for small values of x. Try to figure out, using the hints already given, why must divide .