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Math Help - Eulers theorem

  1. #1
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    Eulers theorem

    Can some1 please help me with this
    made another mistake its
    please help me solve the following without using a calculator

    9^7 mod 143 oops!!

    thanks
    Last edited by darren_a1; April 10th 2006 at 04:09 PM.
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  2. #2
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    Note that 143 = 11.13 so we do it by the Chinese Remainder Theorem, and use Fermat's Little Theorem.

    Modulo 11: 9^7 == (3^2)^7 == 3^14 == 3^10.3^4 == 1.3^4 == 81 == 4

    Modulo 13: 9^7 == (3^2)^7 == 3^14 == 3^12.3^2 == 1.3^2 == 9

    Now to reconstruct the answer. 6.11 - 5.13 = 1 by Euclid's algorithm. So
    6.11.a - 5.13.b is == a mod 13 and == b mod 11. The answer is thus 6.11.9 - 5.13.4 == 594 - 260 == 334 == 48 mod 143.

    Answer: 48.
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