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The order of all elements of (Z/ 7161 Z)* divide 30

  1. May 1, 2012 #1
    Hi,

    1. The problem statement, all variables and given/known data
    Show that for every x in (Z/ 7161 Z)*, the order of x divides 30.

    2. Relevant equations
    (Z/ 7161 Z)* is the group of units of Z/ 7161 Z.


    3. The attempt at a solution

    I factorised 7161: 7161 = 3 * 7 * 11 * 31
    I used the Chinese remainder theorem to show that (Z/ 7161 Z)* has (3-1)*(7-1)*(11-1)*(31-1) = 3600 elements.
    So the order of every x in (Z/ 7161 Z)* has to divide 3600.
    I don't know how to reduce this to 30. Can anyone help me with the next step?

    Thanks.
     
  2. jcsd
  3. May 1, 2012 #2

    I like Serena

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    Homework Helper

    It looks like you've applied Euler's theorem (##\phi## function) instead of the Chinese remainder theorem.

    Chinese remainder theorem says that if p,q,r relatively prime, that then:
    $$(Z/pqrZ)^* \cong (Z/pZ)^* \times (Z/qZ)^* \times (Z/rZ)^*$$

    Can you split your group like this?
    And can you say something about the order of an element x in each of these groups?
     
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