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 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.
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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 ==

Revision as of 19:29, 7 March 2018

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 is 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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