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Math Help - modulo arithmetic help

  1. #1
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    modulo arithmetic help

    I am really having trouble understanding how to do these questions, and I have read up on google, but still struggling


     5^{11} mod   9 = ?

    I need to be able to do this question without a calculator, and i am not sure how to do it? Any help appreciated.
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  2. #2
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    Re: modulo arithmetic help

    Hey Tweety.

    The easiest way to do this is to use powers of 2. Look at 5 mod 11, then 5^2, then 5^4, 5^8, then 5^10 and finally 5^11.

    As an example 5 mod 9 = 5. 5^2 mod 9 = 25 mod 9 = -2. 5^4 = (-2)^2 mod 9 = 4. 5^8 mod 9 = 16 mod 9 = -2. 5^11 mod 9 = -2*-2*5 mod 9 = 20 mod 9 = 2.

    Check to see if I have made a mistake, but the process used above is the basic idea in solving these kinds of problems.
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  3. #3
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    Re: modulo arithmetic help

    Using modular arithmetic of remainder classes: [a][b]=[ab]:

    [5][5]=[25]=[7]
    [5]3=[35]=[8]
    [5]4=[40]=[4]
    [5]5=[20]=[2]
    [5]11=[2][2][5]=[20]=[2]
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  4. #4
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    Re: modulo arithmetic help

    Quote Originally Posted by chiro View Post
    Hey Tweety.

    The easiest way to do this is to use powers of 2. Look at 5 mod 11, then 5^2, then 5^4, 5^8, then 5^10 and finally 5^11.

    As an example 5 mod 9 = 5. 5^2 mod 9 = 25 mod 9 = -2. 5^4 = (-2)^2 mod 9 = 4. 5^8 mod 9 = 16 mod 9 = -2. 5^11 mod 9 = -2*-2*5 mod 9 = 20 mod 9 = 2.

    Check to see if I have made a mistake, but the process used above is the basic idea in solving these kinds of problems.
    Hello,

    Thank you, this is the methos my teacher showed me, but i still couldnt quite get the hang of it,

    please tell me if this working out is correct, thank you.






     5^{1}  mod   9  = 5

     5^{2}  mod 9  = 7

     5^{4}  mod 9 = (5^{2})^{2}  mod 9  = 49 mod 9 = 4

     5^{8} mod 9  = (5^{4}) mod 9 = (4)^{2} mod 9 = 16 mod 9 = 7


    so  5^{11} mod 9 = 5^{1} . 5^{2} . 5^{8} mod 9  = 7 . 7 . 5 = 245 mod 9

    9 x 27 = 243

    so 245/9 = 27

    remainder = 2

    so 245 mod 9 = 2?
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