What 'divides evenly' really means
'3 divides 12' is made precise not by talking about division at all, but multiplication: 12 = 3 × 4. Defining divisibility through multiplication avoids fractions entirely and keeps everything inside the integers, which is what makes clean proofs possible.
Tiles on a floor: 3 divides 12 because 3-inch tiles fit a 12-inch strip exactly, no cutting. 3 does not divide 13 — you'd have to cut a tile.
Does 7 divide 91? Look for k with 91 = 7k. k = 13 works, so yes. Does 7 divide 100? 7×14 = 98 and 7×15 = 105 — no k exists, so no.
Divisibility is the atom of number theory. It underlies remainders, clock arithmetic, hash tables, barcodes, and the RSA encryption protecting your bank login.
Level 1 The precise statement
For integers a, b we say a divides b (written a | b) iff there exists an integer k with b = a * k.
Level 3 What it stands on (1 direct)
- Integers form a commutative ring (axiom)
- Divisibility (definition)
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.
13 downstream results would collapse with it. See the blast radius on the graph →