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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, that 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.
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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, 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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[[Euler's totient function]] determines the number of positive integers less than any given positive integer that are relatively prime to that number.
 
[[Euler's totient function]] determines the number of positive integers less than any given positive integer that are relatively prime to that number.
  
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By the [[Euclidean algorithm]], consecutive positive integers are always relatively prime. This is related to the fact that two numbers <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]]).
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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>.
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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>.
  
 
== See also ==
 
== See also ==

Revision as of 16:25, 16 March 2012

Two positive integers $m$ and $n$ are said to be relatively prime or coprime if they share no common divisors greater than 1, that is their greatest common divisor is $\gcd(m, n) = 1$. Equivalently, $m$ and $n$ must have no prime divisors in common. The positive integers $m$ and $n$ are relatively prime if and only if $\frac{m}{n}$ 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 are relatively prime to that number.

Consecutive positive integers are always relatively prime, since, if a prime $p$ divides both $n$ and $n+1$, then it must divide their difference $(n+1)-n = 1$, which is impossible since $p > 1$.

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

See also

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