Primes can't be fooled by products
Composite numbers CAN be fooled: 6 divides 4×9 = 36, yet 6 divides neither 4 nor 9 (its factors 2 and 3 hid in different places). A prime has no parts to distribute, so if it divides a product it must live wholly inside one factor. This indivisibility of 'blame' is the deep property that makes factorization unique.
A single key that opens a two-room house must open one of the rooms. A keyring (composite) can have its keys split between the rooms.
7 | 84 and 84 = 12 × 7 → 7 | 7 ✓. Try to break it: 7 | (14 × 15) = 210? Yes, and 7 | 14 ✓. Contrast: 6 | 36 = 4×9 but 6∤4 and 6∤9 — composites fail.
This is THE pivotal lemma: uniqueness of prime factorization stands or falls with it, and through that, essentially all of multiplicative number theory.
Level 1 The precise statement
If p is prime and p | (a*b), then p | a or p | b.
Level 2 The proof
Level 3 What it stands on (3 direct)
- Integers form a commutative ring (axiom)
- Divisibility (definition)
- Prime number (definition)
- Well-Ordering Principle (axiom)
- Greatest common divisor (definition)
- Bezout's identity (lemma)
- Euclid's lemma (lemma)
Level 4 The verified record
This page is generated from a machine-checked node. The kernel confirms its dependencies resolve, nothing is circular, and it grounds in axioms (foundation: peano). The content hash below makes tampering evident.
1 downstream result would collapse with it. See the blast radius on the graph →