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Math Help - Fermatís Little Theorem

  1. #1
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    Fermatís Little Theorem

    Use Fermatís Little Theorem to calculate the remainder of 339^8356 when divided by 17.


    339^16 = 1 (mod 17) by Fermat theorems.
    So (raise to the 522) both sides,
    339^8352 = 1 (mod 17)

    Does this mean 339^8352 divided by 17 give remainder 1?

    How do i get the next step?

    339^8352x339^4=339^4(mod 17)

    Whats the next step after this? Please help.

    Thanks
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  2. #2
    Super Member PaulRS's Avatar
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    Just see that 339\equiv{-1}(\bmod{17}) then 339^4\equiv{1}(\bmod{17})

    and so 339^{8356}\equiv{1}(\bmod{17})
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  3. #3
    Super Member PaulRS's Avatar
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    Or better, 8356=<br />
{\dot 4}<br />
since 56=<br />
{\dot 4}<br />
and we have that 339^4\equiv{1}(\bmod.17)

    Thus: 339^{8356}=339^{4\cdot{z}}=(339^{4})^z\equiv{1}(\b  mod.17) were z is the natural such that 8356=4∑z

    Directly 339\equiv{-1}(\bmod.17) so 339^{8356}\equiv{(-1)^{8356}}=1(\bmod.17)
    Last edited by PaulRS; February 25th 2008 at 08:23 AM.
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