Canonical representation?

If $\{p_i\}$ is the set of all prime numbers, we can express $n,m, s$ positive integers as unique products

$n=\prod{p_i^{\alpha_i}}$ , $m=\prod {p_i^{\beta_i}}$ , $s=\prod {p_i^{\gamma_i}}$

where $\alpha_i\geq 0,\beta_i\geq 0,\gamma_i\geq 0$ are finitely many non zero integers.

If $\beta_i\leq \alpha_i$ for all $i$ evidently $m|n$ . If $m|n$ there exists $s$ positive integer such that $n=ms$ that is, $\prod p_i^{\alpha_i}=\prod p_i^{\beta_i+\gamma_i}$ . By the unique factorization, $\alpha_i=\beta_i+\gamma_i$ , hence $\beta_i\leq \alpha_i$ for all $i$ .