# totative

Given a positive integer $n$, an integer $$ is a totative^{} of $n$ if $\mathrm{gcd}(m,n)=1$. 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 |