I'm having trouble with these two questions. Problem 1: Prove that the multiplicative monoid N of natural numbers 1, 2,... is a free commutative monoid. Problem 2: Is the multiplicative monoid N isomorphic to the additive monoid N_0 x N_0 x ...x N_0 (n times), for any n = 1, 2,...? Prove your claim. If i can get any tips on where to begin with them, that would be great. from there i will work on an show where i have progressed. right now, all i can think of is using induction in 2 to prove that there is an isomorphism for all n. i do this, by looking at neutral elements in each monoid, and constructing the isomorphism for n=1. that's all i can think up.