Difference between revisions of "Relatively prime"
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| − | Two [[positive]] [[integer]]s <math>m</math> and <math>n</math> are said to be '''relatively prime''' or '''coprime''' if they share no [[common divisor | common divisors]] greater than 1, | + | Two [[positive]] [[integer]]s <math>m</math> and <math>n</math> are said to be '''relatively prime''' or '''coprime''' if they share no [[common divisor | common divisors]] greater than 1. That is, their [[greatest common divisor]] is <math>\gcd(m, n) = 1</math>. Equivalently, <math>m</math> and <math>n</math> must have no [[prime]] divisors in common. The positive integers <math>m</math> and <math>n</math> are relatively prime if and only if <math>\frac{m}{n}</math> is in lowest terms. |
== Number Theory == | == Number Theory == | ||
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Consecutive positive integers are always relatively prime, since, if a prime <math>p</math> divides both <math>n</math> and <math>n+1</math>, then it must divide their difference <math>(n+1)-n = 1</math>, which is impossible since <math>p > 1</math>. | Consecutive positive integers are always relatively prime, since, if a prime <math>p</math> divides both <math>n</math> and <math>n+1</math>, then it must divide their difference <math>(n+1)-n = 1</math>, which is impossible since <math>p > 1</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 | + | 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>. |
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| + | 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 17:45, 14 October 2024
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
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