You can’t uniquely factor if you include units, which in the naturals is 1. You’d also remove every other prime number, since in general we can say that a prim

You can’t uniquely factor if you include units, which in the naturals is 1. You’d also remove every other prime number, since in general we can say that a prime isn’t divisible by a smaller prime. That we have to add “except 1” to both the factoring theorem and the definition of prime itself is why we don’t include 1 as a prime.