I am asked to prove that the monoids $(\mathbb{N}^*,\cdot)$ and $(3 \mathbb{N} + 1, \cdot)$ are not isomorphic, where $\mathbb{N}^*$ denotes the positive integers (strictly greater than $0$).
I know that an isomorphism $f$ would satisfy $f(1) = 1$ and $f(ab) = f(a)f(b)$ for $a,b \in \mathbb{N}^*$.
My intuition is that in $(3 \mathbb{N} + 1, \cdot)$ there are elements (such as 55) that are prime in this monoid but factorizable in the other.
Since any isomorphism $f$ from $(\mathbb{N}^*,\cdot)$ to $(3 \mathbb{N} + 1, \cdot)$ is a multiplicative function, it is completely defined by its values on the primes in $(\mathbb{N}^*,\cdot)$. I was able to show that $f$ can only map primes of one monoid to primes of the other.
How should I continue?