Difference between revisions of "Greatest common divisor"
From Maths
m |
m |
||
Line 8: | Line 8: | ||
===Co-prime=== | ===Co-prime=== | ||
If for {{M|a,b\in\mathbb{N}_+}} we have {{M|1=\text{gcd}(a,b)=1}} then {{M|a}} and {{M|b}} are said to be ''co-prime''<ref name="Crypt"/> | If for {{M|a,b\in\mathbb{N}_+}} we have {{M|1=\text{gcd}(a,b)=1}} then {{M|a}} and {{M|b}} are said to be ''co-prime''<ref name="Crypt"/> | ||
− | + | ====Coprime==== | |
+ | Coprime is used by some authors, co-prime by others. I prefer the hyphenated form because to be ''co-something'' implies more than one thing is involved. You cannot have "a coprime number" but you can have a pair of co-prime numbers. | ||
==See next== | ==See next== | ||
* [[Euclidean algorithm]] | * [[Euclidean algorithm]] |
Revision as of 08:28, 21 May 2015
Note: requires knowledge of what it means for a number to be a divisor of another.
Definition
Given two positive integers, [ilmath]a,b\in\mathbb{N}_+[/ilmath], the greatest common divisor of [ilmath]a[/ilmath] and [ilmath]b[/ilmath][1] is the greatest positive integer, [ilmath]d[/ilmath], that divides both [ilmath]a[/ilmath] and [ilmath]b[/ilmath]. We write:
- [ilmath]d=\text{gcd}(a,b)[/ilmath]
Terminology
Co-prime
If for [ilmath]a,b\in\mathbb{N}_+[/ilmath] we have [ilmath]\text{gcd}(a,b)=1[/ilmath] then [ilmath]a[/ilmath] and [ilmath]b[/ilmath] are said to be co-prime[1]
Coprime
Coprime is used by some authors, co-prime by others. I prefer the hyphenated form because to be co-something implies more than one thing is involved. You cannot have "a coprime number" but you can have a pair of co-prime numbers.