Axiomatic WikiWWikiDDefinitionsCongruence modulo n
Definition·D113
Two integers are congruent modulo n when their difference is an integer multiple of n.
for
and
, where
is the integer image of
and the difference
uses negation: the difference of
and
is an integer multiple of
.
In words
a is congruent to b modulo n exactly when the difference of a and b is an integer multiple of n.
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 · A09 (computed from the citation graph, not asserted).
Remarks
The familiar reading:
and
leave the same remainder on division by
(T10); equivalently,
divides the difference
in the integers. Two boundary cases:
makes congruence ordinary equality (only
is a multiple of
), and
makes every pair congruent (every integer is a multiple of
). Congruence modulo
is an equivalence relation (L73); indeed it is exactly the coset relation of the subgroup
(L72). It is also compatible with
and
(L74), so it descends to the ring
(D114).
Used by
Propose an edit2 published revisions
- 7/13/2026 · Benjamin· Restate congruence directly as "the difference is an integer multiple of n" instead of citing a separate integer-divisibility article (dropped as redundant with def-divides); fix math-in-citation-label.→what changed →
- 7/13/2026 · Benjamin· Define congruence modulo n on the integers as n dividing the difference, the central relation of modular arithmetic.→what changed →