Definition·D030
Divisibility
One number divides another when it fits in exactly, with no remainder.
for
, with
the multiplication. One writes
for the negation, and calls
a divisor of
.
In words
d divides n exactly when n is d times some natural number.
Rests onno axioms yet
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A02 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
Examples:
with witness
;
, as the division algorithm makes precise (remainder
). Two boundary cases are worth internalizing: every
divides
(witness
), while
divides only
; and
divides everything. The everyday algebra of divisibility is collected in L19; the numbers whose only divisors are forced (
and themselves) are the primes.
Used by
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