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Math Help - solving linear congruences...please help

  1. #1
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    solving linear congruences...please help

    solving linear congruences...please help-congruences.png

    for (e) and (f).... I dont know how to solve such congruences, I only know how to get to an answer when I check for different values of x....

    (g) I can do.

    (h) The answer that was given users Euler's function... can some1 plz show me how to use it for this question....

    any help greatly appreciated..
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  2. #2
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    Quote Originally Posted by Dreamer78692 View Post
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    for (e) and (f).... I dont know how to solve such congruences, I only know how to get to an answer when I check for different values of x....

    (g) I can do.

    (h) The answer that was given users Euler's function... can some1 plz show me how to use it for this question....

    any help greatly appreciated..

    You'll likely need FLT = Fermat's Last Theorem (google it if you don't know it):

    (e) as 2^5=-1\!\!\pmod {11}\,,\,2 is a generator of the multiplicative group \left(\mathbb{Z}/11\mathbb{Z}\right)^* , and thus there does exist some

    unique power of it that equals 7 modulo 11. Try and error (since discrete logarithms seem to be out of our depth for

    the time being) gives us x=7

    (f) Do a list of quadratic residues modulo 11, and check which one, if any, equals 9^x=(-2)^x\!\!\pmod {11} ...

    (h) It's easy to check that 5^6=1\!\!\pmod 7\,,\,\,5^5=1\!\!\pmod {11}\Longrightarrow 5^{60}=1\!\!\pmod {77} (why?)

    In fact, try to show that already 5^{30}=1\!\!\pmod{77} ...

    Tonio
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