Coprime

From Academic Kids

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

Properties

The numbers a and b are coprime if and only if there exist integers x and y such that ax + by = 1 (see Bzout's identity). Equivalently, b has a multiplicative inverse modulo a: there exists an integer y such that by ≡ 1 (mod a). In fancy language: a and b are coprime if and only if b yields a unit in the ring Za of integers modulo a.

As a consequence, if a and b are coprime and brbs (mod a), then rs (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.

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 coprime is 6/π2 (see pi), which is about 60%.

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

Generalizations

Two ideals A and B in the commutative ring R are called coprime if A + B = R. This generalizes Bzout'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 = AB; 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, .... an}) to mean that the greatest common divisor of the elements of the set is 1. If there does not exist any relatively prime pair of integers in the set, then the set is called pairwise relatively prime.

See also

de:Teilerfremdheit fr:Nombres premiers entre eux he:מספרים זרים it:Coprimo ko:서로 소 nl:Relatief priem ru:Взаимно простые числа sl:Tuje število th:จำนวนเฉพาะสัมพัทธ์ zh:互質

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