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 divisor | common divisors]], so <math>\displaystyle\gcd(m,n)=1</math>. Alternatively, <math>{m}</math> and <math>{n}</math> must have no [[prime | + | Two '''relatively prime''' integers <math>{m}</math> and <math>{n}</math> share no [[common divisor | common divisors]] greater than 1, so <math>\displaystyle\gcd(m,n)=1</math>. Alternatively, <math>{m}</math> and <math>{n}</math> must have no [[prime]] divisors 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. | ||
Revision as of 11:24, 20 June 2006
(Also called coprime.)
Two relatively prime integers 
and 
share no common divisors greater than 1, so 
. Alternatively, and must have no prime divisors 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.
It is also worth noting that by the Euclidean algorithm, consecutive positive integers are always relatively prime.
