Coprime

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In mathematics, the integers a and b are said to be coprime or relatively prime if 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. The number 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.

Euler's totient function (or Euler's phi function) of a positive integer n is the number of integers between 1 and n − 1 which are coprime to n.

Contents 1 Properties 2 Cross notation, group 3 Generalizations 4 See also

Properties

Image:Coprime-lattice.svg

There are a number of conditions which are equivalent to a and b being coprime:

• There exist integers x and y such that ax + by = 1 (see Bézout's identity).
• The integer b has a multiplicative inverse modulo a: there exists an integer y such that by ≡ 1 (mod a). In other words, b is a unit in the ring Z/aZ of integers modulo a.

As a consequence, if a and b are coprime and br ≡ bs (mod a), then r ≡ s (mod a) (because we may "divide by b" when working modulo a). Furthermore, if a and b₁ are coprime, and a and b₂ are coprime, then a and b₁b₂ are also coprime (because the product of units is a unit).

If a and b are coprime and a divides a product bc, then a divides c. This can be viewed as a generalisation of Euclid's lemma, which states that if p is prime, and p divides a product bc, then either p divides b or p divides c.

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). (See figure 1.)

The probability that two randomly chosen integers are coprime is 6/π² (see pi), which is about 60%.

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

Cross notation, group

If n≥1 is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation; it is written as (Z/nZ)^× or Z_n^*.

Generalizations

Two ideals A and B in the commutative ring R are called coprime if A + B = R. This generalizes Bézout's identity: 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.

If the ideals A and B of R are coprime, then AB = A∩B; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese Remainder Theorem is an important statement about coprime ideals.

The concept of being relatively prime can also be extended any finite set of integers S = {a₁, a₂, .... a_n} to mean that the greatest common divisor of the elements of the set is 1. If every pair of integers in the set is relatively prime, then the set is called pairwise relatively prime.

Every pairwise relatively prime set is relatively prime; however, the converse is not true: {6, 10, 15} is relatively prime, but not pairwise relative prime. (In fact, each pair of integers in the set has a non-trivial common factor.)

See also

• Greatest common divisorbg:Взаимно прости числа

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