# Difference between revisions of "Relatively prime"

m |
m |
||

Line 10: | Line 10: | ||

Two integers <math>a</math> and <math>b</math> are relatively prime if and only if there exist some <math>x,y\in \mathbb{Z}</math> such that <math>ax+by=1</math> (a special case of [[Bezout's Lemma]]). The [[Euclidean Algorithm]] can be used to compute the coefficients <math>x,y</math>. | Two integers <math>a</math> and <math>b</math> are relatively prime if and only if there exist some <math>x,y\in \mathbb{Z}</math> such that <math>ax+by=1</math> (a special case of [[Bezout's Lemma]]). The [[Euclidean Algorithm]] can be used to compute the coefficients <math>x,y</math>. | ||

− | For two relatively prime numbers, their [[common | + | For two relatively prime numbers, their [[least common multiple]] is their product. This pops up in [[Chinese Remainder Theorem]]. |

== See also == | == See also == |

## Latest revision as of 15:51, 25 February 2020

Two positive integers and are said to be **relatively prime** or **coprime** if they share no common divisors greater than 1. That is, their greatest common divisor is . Equivalently, and must have no prime divisors in common. The positive integers and are relatively prime if and only if is in lowest terms.

## Number Theory

Relatively prime numbers show up frequently in number theory formulas and derivations:

Euler's totient function determines the number of positive integers less than any given positive integer that is relatively prime to that number.

Consecutive positive integers are always relatively prime, since, if a prime divides both and , then it must divide their difference , which is impossible since .

Two integers and are relatively prime if and only if there exist some such that (a special case of Bezout's Lemma). The Euclidean Algorithm can be used to compute the coefficients .

For two relatively prime numbers, their least common multiple is their product. This pops up in Chinese Remainder Theorem.

## See also

*This article is a stub. Help us out by expanding it.*