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Math Help - proving an equation using congruences

  1. #1
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    Feb 2010
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    proving an equation using congruences

    prove that 11 divides 456^654 + 123^321.

    I know that you can turn it into a congruence.

    456^654 is congruent to -123^321 (mod 11)

    But then I have no idea where to go.
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  2. #2
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    Joined
    Aug 2009
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    \phi(11)=10
    456^{10}\equiv 1\,mod\,11
    456^{650}\equiv 1\,mod\,11
    456\equiv 5\,mod\,11
    456^4\equiv5^4\equiv625\equiv9\,mod\,11
    456^{650}\cdot 456^4\equiv 1\cdot 9\equiv 9\,mod\,11
    same here
    123^{320}\equiv 1\,mod\,11
    123^{320}\cdot 123 \equiv 1\cdot 123\equiv 2\,mod\,11
    so
    456^{654}+123^{321}\equiv 2+9\equiv 0\,mod\,11
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