Difference between revisions of "Greatest common divisor"
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==Definition== | ==Definition== | ||
| − | {{ | + | Given two positive integers, {{M|a,b\in\mathbb{N}_+}}, the ''greatest common divisor'' of {{M|a}} and {{M|b}}<ref name="Crypt">The mathematics of ciphers, Number theory and RSA cryptography - S. C. Coutinho</ref> is the greatest positive integer, {{M|d}}, that [[Divisor|divides]] both {{M|a}} and {{M|b}}. We write: |
| + | * {{M|1=d=\text{gcd}(a,b)}} | ||
| + | |||
| + | ==Terminology== | ||
| + | ===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"/> | ||
| + | |||
| + | ==See next== | ||
| + | * [[Euclidean algorithm]] | ||
| + | |||
| + | ==See also== | ||
| + | * [[Division algorithm]] | ||
| + | * [[Divisor]] | ||
| + | |||
| + | ==References== | ||
| + | <references/> | ||
{{Definition|Number Theory}} | {{Definition|Number Theory}} | ||
Revision as of 08:25, 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]
