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Thread: divisibility proof

  1. December 2nd 2007, 10:02 AM #1
    frankdent1
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    divisibility proof

    How would I go about proving 165 | (n^20 - a^20) if n and a are relatively prime to 165?
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  2. December 2nd 2007, 10:08 AM #2
    PaulRS
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    First: 165=3\cdot{5}\cdot{11}

    By Fermat's Little Theorem n^{4}\equiv{1}(\bmod.5) (since n is coprime to 5) then n^{20}\equiv{1^5=1}(\bmod.5) and in the same way: a^{20}\equiv{1}(\bmod.5)

    THus: n^{20}-a^{20}\equiv{0}(\bmod.5) so 5 divides n^{20}-a^{20}

    Do the same for the other two primes ( 3 and 11) and you have that 165 divides n^{20}-a^{20}
    Last edited by PaulRS; December 2nd 2007 at 11:15 AM. Reason: Latex
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  3. December 2nd 2007, 10:39 AM #3
    ThePerfectHacker
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    Paul use \bmod instead.
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