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When is m33 divisible by 36? (m is an integer).

  1. Jan 6, 2012 #1
    1. The problem statement, all variables and given/known data

    Work out the order of the following elements;

    [itex]33 \in Z_{36}[/itex]


    3. The attempt at a solution

    It's probably really simple. But this only happens when an integer times 33 is divisible by 36.

    That is;

    [itex] 33n = 36m [/itex]

    Which I can re-arrange to find

    [itex] n = 36m/33 [/itex]

    Now, I can keep adding 36/33 in my calculator until I get an integer result, but surely there is an easier way? For example;

    [itex] 33 \equiv -3mod36 [/itex]

    Which suggests that 36/3 = 12 is the order of 33. This is fine and dandy, but what happens when I get to a question like, find the order of

    [itex]15 \in Z_{36}[/itex]

    Then

    [itex] 15 \equiv -21mod36 [/itex]

    Which means I'm back to the same problem again. My modular arithmetic is fairly poor, so how would I work this out?
     
  2. jcsd
  3. Jan 6, 2012 #2

    LCKurtz

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    Factor 36 into prime factors and think about what factors k must have for 36 to divide 33k.
     
  4. Jan 6, 2012 #3
    36 = 3x3x2x2

    33 = 11x3

    So k must be 3x2x2 = 12.

    Also

    15 = 5x3

    So k must be 3x2x2 = 12.


    Thanks!
     
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