If two numbers b and c have the property that their difference is integrally divisible by a number m (i.e., is an integer), then b and c are said to be "congruent modulo m." The number m is called the modulus, and the statement "b is congruent to c (modulo m)" is written mathematically as

If is not integrally divisible by m, then we say "b is not congruent to c (modulo m)," which is written

The explicit "(mod m)" is sometimes omitted when the modulus m is understood by context, so in such cases, care must be taken not to confuse the symbol with the equivalence sign.

The quantity b is sometimes called the "base," and the quantity c is called the residue or remainder. There are several types of residues. The common residue defined to be nonnegative and smaller than m, while the minimal residue is c or , whichever is smaller in absolute value.

Congruence arithmetic is perhaps most familiar as a generalization of the arithmetic of the clock. Since there are 60 minutes in an hour, "minute arithmetic" uses a modulus of m = 60. If one starts at 40 minutes past the hour and then waits another 35 minutes, , so the current time would be 15 minutes past the (next) hour.

Similarly, "hour arithmetic" on a 12-hour clock uses a modulus of m = 12, so 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: (which can be regarded as a definition).

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 (k, m) is the greatest common divisor.

13. If , then , for P(x) a polynomial.

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

where r and s are integers. Then
so the properties are true.

Congruences also apply to fractions. For example, note that

so

To find (mod m) where (i.e., q and m are relatively prime), use an algorithm similar to the greedy algorithm. Let and find

where is the ceiling function, then compute

Iterate until , then

This method always works for m prime, and sometimes even for m composite. However, for a composite m, the method can fail by reaching 0 (Conway and Guy 1996).

Finding a fractional congruence is equivalent to solving a corresponding linear congruence equation

A fractional congruence of a unit fraction is known as a modular inverse. A fractional congruence can be found in Mathematica using the following function:

MyMod[r_Rational, m_Integer] := Mod[
Numerator[r]PowerMod[Denominator[r], -1, m], m]
MyMod[i_, m_] := Mod[i, m]
      

or using the undocumented syntax PolynomialMod[r, m].

Algebraic Congruence, Cancellation Law, Chinese Remainder Theorem, Common Residue, Congruence Axioms, Congruence Equation, Congruent, Divisibility Tests, Floor Function, Fractional Part, Functional Congruence, Geometric Congruence, Greatest Common Divisor, Integer Division, Least Common Multiple, Linear Congruence Equation, Minimal Residue, Mod, Modular Inverse, Modulus, Nearest Integer Function, Quadratic Congruence Equation, Quadratic Reciprocity Law, Quotient, Residue, RSA Encryption