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/π^{2} (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.

See also: Greatest common divisor