# Euler phi at a product

If the positive greatest common divisor^{} of the integers $a$ and
$b$ is $d$, then

$$\phi (ab)=\frac{\phi (a)\phi (b)d}{\phi (d)}.$$ |

*Proof.* Using the positive prime factors^{} $p$, the right hand side of the asserted equation is

$\frac{d\cdot a{\prod}_{p\mid a}\frac{p-1}{p}\cdot b{\prod}_{p\mid b}\frac{p-1}{p}}{d{\prod}_{p\mid a,p\mid b}\frac{p-1}{p}}$ | $\mathrm{\hspace{0.33em}}={\displaystyle \frac{ab{\prod}_{p\mid a,p\nmid b}\frac{p-1}{p}\cdot {\prod}_{p\mid a,p\mid b}\frac{p-1}{p}\cdot {\prod}_{p\mid b,p\nmid a}\frac{p-1}{p}\cdot {\prod}_{p\mid b,p\mid a}\frac{p-1}{p}}{{\prod}_{p\mid a,p\mid b}\frac{p-1}{p}}}$ | ||

$\mathrm{\hspace{0.33em}}=ab{\displaystyle \prod _{p\mid a\vee p\mid b}}{\displaystyle \frac{p-1}{p}}=ab{\displaystyle \prod _{p\mid ab}}{\displaystyle \frac{p-1}{p}}=\phi (ab),$ |

Q.E.D.

Title | Euler phi at a product^{} |
---|---|

Canonical name | EulerPhiAtAProduct |

Date of creation | 2014-02-18 14:02:24 |

Last modified on | 2014-02-18 14:02:24 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 6 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 11A25 |

Classification | msc 11-00 |

Related topic | EulerPhifunction |

Related topic | DivisibilityByPrimeNumber |