# Difference between revisions of "Relatively prime"

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(Also called ''coprime''.) | (Also called ''coprime''.) | ||

− | Two '''relatively prime''' integers <math>{m}</math> and <math>{n}</math> share no common factors. Alternatively, <math>{m}</math> and <math>{n}</math> must have no [[prime factor|prime factors]] in common. For example, 15 and 14 are relatively prime; as the [[prime factorization]] of 15 is <math>3 \cdot 5</math>, the prime factorization of 14 is <math>2 \cdot 7</math>, and no prime factors are shared between the two. | + | Two '''relatively prime''' integers <math>{m}</math> and <math>{n}</math> share no common factors, so their <math>\gcd</math> is <math>1</math>. Alternatively, <math>{m}</math> and <math>{n}</math> must have no [[prime factor|prime factors]] in common. For example, 15 and 14 are relatively prime; as the [[prime factorization]] of 15 is <math>3 \cdot 5</math>, the prime factorization of 14 is <math>2 \cdot 7</math>, and no prime factors are shared between the two. |

Note that for relatively prime <math>{m}</math> and <math>{n}</math>, <math>\frac{m}{n}</math> will be in lowest terms. | Note that for relatively prime <math>{m}</math> and <math>{n}</math>, <math>\frac{m}{n}</math> will be in lowest terms. | ||

− | Relatively prime numbers show up frequently in [[number theory]] formulas and derivations. [[Euler's totient function]], for example, determines the number of positive integers less than any given positive integer that are relatively prime to that number. The [[Chicken mcNugget theorem]] also involves | + | Relatively prime numbers show up frequently in [[number theory]] formulas and derivations. [[Euler's totient function]], for example, determines the number of positive integers less than any given positive integer that are relatively prime to that number. The [[Chicken mcNugget theorem]] also involves relatively prime numbers. |

## Revision as of 04:13, 20 June 2006

(Also called *coprime*.)

Two **relatively prime** integers and share no common factors, so their is . Alternatively, and must have no prime factors in common. For example, 15 and 14 are relatively prime; as the prime factorization of 15 is , the prime factorization of 14 is , and no prime factors are shared between the two.

Note that for relatively prime and , will be in lowest terms.

Relatively prime numbers show up frequently in number theory formulas and derivations. Euler's totient function, for example, determines the number of positive integers less than any given positive integer that are relatively prime to that number. The Chicken mcNugget theorem also involves relatively prime numbers.