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Thread: divisibility proof

  1. #1
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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. #2
    Super Member PaulRS's Avatar
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    First: $\displaystyle 165=3\cdot{5}\cdot{11}$

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

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

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