If is integrally divisible by , then and are said to be congruent with Modulus . This is written mathematically as (mod ). If is *not* divisible by , then we
say (mod ). The (mod ) is sometimes omitted when the Modulus is
understood for a given computation, so care must be taken not to confuse the symbol with that for an
Equivalence. The quantity is called the Residue or
Remainder. The Common Residue is taken to be Nonnegative and smaller than , and the Minimal
Residue is or , whichever is smaller in Absolute Value. In many computer languages (such as `FORTRAN` or
*Mathematica*
), the Common Residue of (mod ) is written `mod(c,a)`.

Congruence arithmetic is perhaps most familiar as a generalization of the arithmetic of the clock: 40 minutes past the hour plus 35 minutes gives , or 15 minutes past the hour, and 10 o'clock a.m. plus five hours gives , or 3 o'clock p.m. Congruences satisfy a number of important properties, and are extremely useful in many areas of Number Theory. Using congruences, simple Divisibility Tests to check whether a given number is divisible by another number can sometimes be derived. For example, if the sum of a number's digits is divisible by 3 (9), then the original number is divisible by 3 (9).

Congruences also have their limitations. For example, if and , then it follows that , but usually not that or . In addition, by ``rolling over,'' congruences discard absolute information. For example, knowing the number of minutes past the hour is useful, but knowing the hour the minutes are past is often more useful still.

Let and , then important properties of congruences include the following, where means ``Implies'':

- 1. Equivalence: .
- 2. Determination: either or .
- 3. Reflexivity: .
- 4. Symmetry: .
- 5. Transitivity: and .
- 6. .
- 7. .
- 8. .
- 9. .
- 10. .
- 11. and , where is the Least Common Multiple.
- 12. , where is the Greatest Common Divisor.
- 13. If , then , for a Polynomial.

Properties (6-8) can be proved simply by defining

(1) | |||

(2) |

where and are Integers. Then

(3) | |||

(4) | |||

(5) |

so the properties are true.

Congruences also apply to Fractions. For example, note that

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

A general Quadratic Congruence

(13) |

(14) |

(15) |

Two simultaneous congruences

(16) |

(17) |

(18) |

**References**

Conway, J. H. and Guy, R. K. ``Arithmetic Modulo .'' In *The Book of Numbers.* New York: Springer-Verlag,
pp. 130-132, 1996.

Courant, R. and Robbins, H. ``Congruences.'' §2 in Supplement to Ch. 1 in
*What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.*
Oxford, England: Oxford University Press, pp. 31-40, 1996.

Shanks, D. *Solved and Unsolved Problems in Number Theory, 4th ed.* New York: Chelsea, p. 55, 1993.

Weisstein, E. W. ``Fractional Congruences.'' Mathematica notebook ModFraction.m.

© 1996-9

1999-05-26