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congruence (Definition)

Let $ a$, $ b$ be integers and $ m$ a non-zero integer. We say that $ a$ is congruent to $ b$ modulo $ m$, if $ m$ divides $ b-a$ (the word modulo is the dative case of the Latin noun modulus meaning the 'module'). We write this number congruence or shortly congruence as

$\displaystyle a\equiv b\pmod{m}.$

If $ a$ and $ b$ are congruent modulo $ m$, it means that both numbers leave the same residue when divided by $ m$.

Congruence with a fixed module is an equivalence relation on $ \mathbb{Z}$. The set of equivalence classes, the so-called residue classes, is a cyclic group of order $ m$ (assuming it positive) with respect to addition and a ring if we consider also the multiplication modulo $ m$. This ring is usually denoted as

$\displaystyle \frac{\mathbb{Z}}{m\mathbb{Z}}$
and called the residue class ring modulo $ m$. This ring is also commonly denoted as $ \mathbb{Z}_m$, $ \mathbb{Z}/(m)$. However, when $ m = p$ is a prime number, notation $ \mathbb{Z}_p$ is also used to denote $ p$-adic numbers.

Properties of congruences

  1. If $ a\equiv b\pmod{m}$, then $ a\!+\!c\equiv b\!+\!c\pmod{m}$ and $ ac\equiv bc\pmod{m}$.
  2. If $ a\equiv b\pmod{m}$ and $ c\equiv d\pmod{m}$, then $ a\!\pm\!c\equiv b\!\pm\!d\pmod{m}$ and $ ac\equiv bd\pmod{m}$.
  3. If $ a\equiv b\pmod{m}$ and $ f$ is a polynomial with integer coefficients, then $ f(a)\equiv f(b)\pmod{m}$.
  4. If $ ac\equiv bc\pmod{m}$ and $ \gcd(c,\,m) = 1$, then $ a\equiv b\pmod{m}$.
  5. If $ ac \equiv bc \pmod{m}$, then $ a \equiv b \pmod{\frac{m}{\gcd(c,\,m)}}$.

Proof of 5. Let $ \gcd(c,\,m) := d,\;\; c := c'd,\;\; m := m'd$, where $ \gcd(c',\,m') = 1$. The given congruence means that $ m \mid (a\!-\!b)c$, whence $ m' \mid (a\!-\!b)c'$. Since $ c'$ and $ m'$ are coprime, we infer that $ m' \mid a\!-\!b$, i.e. $ a \equiv b \pmod{m'}$. Q.E.D.



"congruence" is owned by rspuzio. [ full author list (3) | owner history (2) ]
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See Also: congruence, gcd, example of gcd, ${\mathbb{Z}}_n$, ideal norm

Other names:  congruent
Also defines:  number congruence, residue class, residue class ring
Keywords:  number theory, integers, divisibility

Attachments:
linear congruence (Definition) by Mathprof
polynomial congruence (Definition) by pahio
prime residue class (Definition) by pahio
residue systems (Definition) by pahio
quadratic congruence (Theorem) by pahio
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Cross-references: coprime, coefficients, polynomial, prime number, multiplication, ring, addition, positive, order, cyclic group, equivalence classes, equivalence relation, fixed, residue, module, divides, integers
There are 77 references to this entry.

This is version 13 of congruence, born on 2001-10-06, modified 2008-01-27.
Object id is 101, canonical name is Congruences.
Accessed 15656 times total.

Classification:
AMS MSC: 11A05 (Number theory :: Elementary number theory :: Multiplicative structure; Euclidean algorithm; greatest common divisors)
 11A07 (Number theory :: Elementary number theory :: Congruences; primitive roots; residue systems)
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