Divisible iff PFEL is less than or equal

Dependencies:

  1. Prime Factorization Exponent List (PFEL)
  2. PFEL of ratio is difference of PFEL

$b \mid a \iff \operatorname{PFEL}(a) \ge \operatorname{PFEL}(b)$

Proof

Let $\operatorname{PFEL}(a)_i = a_i$ and $\operatorname{PFEL}(b)_i = b_i$.

\begin{align} & b \mid a \\ &\iff \frac{a}{b} \in \mathbb{Z} \\ &\iff \forall i \in \mathbb{N}, \operatorname{PFEL}(\frac{a}{b})_i \ge 0 \\ &\iff \forall i \in \mathbb{N}, a_i - b_i \ge 0 \\ &\iff \forall i \in \mathbb{N}, \operatorname{PFEL}(a)_i \ge \operatorname{PFEL}(b)_i \\ &\iff \operatorname{PFEL}(a) \ge \operatorname{PFEL}(b) \end{align}

Dependency for:

  1. PFEL of gcd is min of PFEL
  2. PFEL of lcm is max of PFEL

Info:

Transitive dependencies:

  1. Every number has a prime factorization
  2. Integer Division Theorem
  3. GCD is the smallest Linear Combination
  4. Euclid's lemma
  5. Fundamental Theorem of Arithmetic
  6. Prime Factorization Exponent List (PFEL)
  7. PFEL of ratio is difference of PFEL