Coprime

In mathematics, the integers a and b are said to be coprime or relatively prime iff they have no common factor other than 1 and -1, or equivalently, if their greatest common divisor is 1.

For example, 6 and 35 are coprime, but 6 and 27 are not because they are both divisible by 3. 1 is coprime to every integer; 0 is coprime only to 1 and -1.

A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm.

Properties

The numbers a and b are coprime if and only if there exist integers x and y such that ax + by = 1 (see Bézout's identity). Equivalently, b has a multiplicative inverse modulo a: there exists an integer y such that by ≡ 1 (mod a).

If a and b are coprime and a divides a product bc, then a divides c.

If a and b are coprime and bx ≡ by (mod a), then x ≡ y (mod a). In other words: b yields a unit in the ring Z_a of integers modulo a.

The two integers a and b are coprime if and only if the point with coordinates (a,b) in an Cartesian coordinate system is "visible" from the origin (0,0), in the sense that there is no point with integer coordinates between the origin and (a,b).

The probability that two randomly chosen integers are relatively prime is 6/π² (see Pi).

Two natural numbers a and b are coprime if and only if the numbers 2^a-1 and 2^b-1 are coprime.

Generalization

Two ideals A and B in the commutative ring R are called coprime if A + B = R. This generalizes Bezout's identity. If A and B are coprime, then AB = A∩B; furthermore, if C is a third ideal such that A contains BC, then A contains C.

With this definition, two principal ideals (a) and (b) in the ring of integers Z are coprime if and only if a and b are coprime.

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See also: Greatest common divisor