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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:
If a, b, c and d are any integers with a b (mod m) and c d (mod m), then
See Also: Residue