The SlugMath Wiki is under heavy development!
State/Divisibility corresponds to inequalities of prime exponents
From SlugmathWiki
Proposition: (Divisibility corresponds to inequalities of prime exponents) Suppose that $a,b \in \NN$. Let $P$ be the set of prime numbers. Let $(e_p)$ and $(f_p)$ be the exponents in the canonical decomposition of $a,b$ into primes: $$a = \prod_{p \in P} p^{e_p}, \mbox{ and } b = \prod_{p \in P} p^{f_p}.$$
Then $a$ divides $b$ if and only if $e_p \leq f_p$ for all $p \in P$.
Logical Connections
This statement logically relies on the following definitions and statements: Def/Canonical decomposition into primes
The following statements and definitions rely on the material of this page: State/Canonical decompositions can be used to find GCD and LCM
To visualize the logical connections between this statements and other items of mathematical knowledge, you can visit the following cluster(s), and click the "Visualize" tab: Clust/Basic number theory
