Greatest common divisor
Note: requires knowledge of what it means for a number to be a divisor of another.
Contents
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]
Proper definition - UNCONFIRMED
This section is Alec's own speculation - not yet verified
Using the Well-ordered principle (given the set of divisors is a finite set, the set has a maximum element, and the maximum is the same as the [ilmath]\text{Sup} [/ilmath] of the set) we can state the [ilmath]\text{gcd} [/ilmath] as follows:
- [math]\text{gcd}(a,b)=\text{sup}(\{n\in\mathbb{N}|n\text{ divides } a\}\cap\{n\in\mathbb{N}|n\text{ divides } b\})[/math]
End of the unconfirmed part
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.
