Prime-exponent parametrization of smooth numbers ($p_1^{e_1}\cdots p_r^{e_r}⟷(e_1,\dots ,e_r)$)

For distinct primes $p_1,\dots ,p_r$, let $M$ consist of the positive integers having no other prime factors, including $1$. The map from a number to its exponent tuple is a bijection from $M$ to ${\mathbb{N}}^r$ when $0\in \mathbb{N}$. The fundamental theorem of arithmetic proves that the tuple is uniquely defined and that equality of tuples implies equality of numbers. Every nonnegative tuple gives the allowed number $p_1^{e_1}\cdots p_r^{e_r}$, proving surjectivity. Multiplication of numbers corresponds to coordinatewise addition of tuples, with $1$ corresponding to the zero tuple.