If two numbers and have the property that their difference is integrally divisible by a number (i.e., is an integer), then and are said to be "congruent modulo ." The number is called the modulus, and the statement " is congruent to (modulo )" is written mathematically as
If is not integrally divisible by , then it is said that " is not congruent to (modulo )," which is written
The explicit "(mod )" is sometimes omitted when the modulus is understood by context, so in such cases, care must be taken not to confuse the symbol with the equivalence sign.
The quantity is sometimes called the "base," and the quantity is called the residue or remainder. There are several types of residues. The common residue defined to be nonnegative and smaller than , while the minimal residue is 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 . 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 , 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 .
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
where and are integers. Then
so the properties are true.
Congruences also apply to fractions. For example, notethat
To find (mod ) where (i.e., and 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 prime, and sometimes even for composite. However, for a composite , the method can fail by reaching 0 (Conway and Guy 1996).
Finding a fractional congruence is equivalent to solving a corresponding linearcongruence equation
A fractional congruence of a unit fraction is known as a modular inverse. A fractional congruence can be found in the Wolfram Language using the following function:
FractionalMod[r_Rational, m_Integer] := Mod[ Numerator[r]PowerMod[Denominator[r], -1, m], m]
or using the undocumented syntax PolynomialMod[r, m] for an explicit rational number.