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Math Help - using fermat's little theorum and problem solving

  1. #1
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    using fermat's little theorum and problem solving

    Give two different methods for finding 2^999(mod 5), one using Fermat's little theorem and the other using basic problem solving methods.
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  2. #2
    Senior Member roninpro's Avatar
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    I'll give you a hint for a basic method. Write 2^{999}\equiv 2\cdot 2^{998}\equiv 2\cdot 4^{499}\pmod{5}. Can you compute this directly?
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  3. #3
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    Quote Originally Posted by apple2010 View Post
    Give two different methods for finding 2^999(mod 5), one using Fermat's little theorem and the other using basic problem solving methods.
    Using Fermats little theorem  2^4 \equiv 1 \pmod{5}
    Then  (2^4)^{249} \equiv 1 \pmod{5}
    Finally  (2^4)^{249}*2^3 \equiv 1*2^3 \pmod{5}
    therefore  2^3 \equiv 3 \pmod{5}
    So  2^{999} \equiv 3 \pmod{5}
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