I know I'm missing something, but I don't see how it was actaully proven. The proof given forces me to conclude only this: that if a prime p dvides the left hand side, then it divides the right hand side--and vice versa. But this is not the same as the left hand side equals the right hand side, which is what needs to be proven. Could someone explain why the proof works?

When p is composite, it could be factored (not uniquely) into divisors of N_i (and, naturally, M). For every prime divisor q of p there is at least one i such that N_i is divisible by the highest power of q in lcm. So, for every i, N_i is associated with the product of maximum powers of q's.

Something's along these lines.