congruence (another Prime Pages' Glossary entries)
 Glossary: Prime Pages: Top 5000: One of the most important tools in elementary number theory is modular arithmetic (or congruences). Suppose a, b and m are any integers with m not zero, then we say a is congruent to b modulo m if m divides a-b. We write this as a b (mod m). For example: 6 2 (mod 4), -1 9 (mod 5), 1100 2 (mod 9), and the square of any odd number is 1 modulo 8. Congruences are found throughout our lives. For example, clocks work either modulo 12 or 24 for hours, and modulo 60 for minutes and seconds. Calendars work modulo 7 for days of the week and modulo 12 for months. The language of congruences was developed by Carl Friedrich Gauss in the early nineteenth century. Notice a b (mod m) if and only if there is an integer q such that a = b + qm, so congruences can be translated to equalities with the addition of one unknown. Perhaps the three most important properties of congruences modulo m are: The reflexive property: If a is any integer, a a (mod m), The symmetric property: If a b (mod m), then b a (mod m), The transitive property: If a b (mod m) and b c (mod m), then a c (mod m). Because of these three properties, we know the set of integers is divided into m different congruence classes modulo m. If a, b, c and d are any integers with a b (mod m) and c d (mod m), then a + c b + d (mod m) a - c b - d (mod m) ac bd (mod m) If gcd(c,m) 1 and ac bc (mod m), then a b (mod m) See Also: Residue Chris K. Caldwell © 1999-2018 (all rights reserved)