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Math Help - Is this the correct...?

  1. #1
    Junior Member universalsandbox's Avatar
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    Is this the correct...?

    Is this the correct cayley table for the units of integers mod 21?




    how would you find the order of each element?
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  2. #2
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    Quote Originally Posted by universalsandbox View Post
    how would you find the order of each element?
    What ye did looks correct.

    To find the order of, say [5]_{21} you need to take exponents until you reach [1]_{21}.

    [5]_{21}^2 = [4]_{21}, [5]_{21}^3 = [20]_{21}, [5]_{21}^4 = [16]_{21}, [5]_{21}^5 = [17]_{21}, [5]_{21}^6 = [1]_{21}.

    Therefore the order of [5]_{21} is 6.
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  3. #3
    Junior Member universalsandbox's Avatar
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    I think I've got it now.

    <br />
[16]_{21} : <br />
[16]_{21}^2 = [4]_{21}, <br />
[16]_{21}^3 = [1]_{21}<br />

    Order 3

    This seems very tedious. Is there a faster way (by hand)?
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  4. #4
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    Quote Originally Posted by universalsandbox View Post
    This seems very tedious. Is there a faster way (by hand)?
    Yes, we can show the order of an element must divide \phi(n). In this case \phi(20) = 8.

    Therefore, the only exponent you need to check are 2,4,8. Instead of all of them.
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  5. #5
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    I believe in this case

    phi(21) = 12

    Note: There are 12 units.

    So the orders must be 1, 2, 3, 4, 6 or 12.

    Ponder this statement: To find the order of the element it is suffice to check only it's 2nd and 3rd power.

    Why?
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