totative

Given a positive integer $n$, an integer is a totative of $n$ if $\gcd (m,n)=1$. Put another way, all the smaller integers than $n$ that are coprime to $n$ are totatives of $n$.

For example, the totatives of 21 are 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19 and 20.

The count of totatives of $n$ is Euler’s totient function $\varphi (n)$. The set of totatives of $n$ forms a reduced residue system modulo $n$. The word “totative” was coined by James Joseph Sylvester, who also coined “totient” (though despite occasional usage in some papers and books, the term “totative” has not caught on the way “totient” has).

Title	totative
Canonical name	Totative
Date of creation	2013-03-22 16:58:16
Last modified on	2013-03-22 16:58:16
Owner	CompositeFan (12809)
Last modified by	CompositeFan (12809)
Numerical id	9
Author	CompositeFan (12809)
Entry type	Definition
Classification	msc 11A25
Related topic	ResidueSystems