Oh god. Do you actually have to exhibit the surjection? You could note that the map n,m)\mapsto 2^n3^m" alt="f:\mathbb{N}\times\mathbb{N}\to\mathbb{N}n,m)\mapsto 2^n3^m" /> is an injection and conclude that there must be a surjection the other way.

Or, I think is a bijection, I think.