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Math Help - Find the remainder

  1. #1
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    Find the remainder

    Find the remainder when (144^{255}+192^{255}) is divided by 49
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  2. #2
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    i tried using eulers theorem but cudnt proceed
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  3. #3
    Moo
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    Hello,

    Note that 3*49=147, and thus 144=-3 mod 49.
    And 4*49=196 ---> 192=-4 mod 49

    So you're left to compute -3^{255}-4^{255} (\bmod 49)

    But \varphi(49)=6\times 7=42

    And 255=42\times 6+3

    So 3^{255}=(3^{42})^6\times 3^3\equiv 3^3 (\bmod 49)

    Similarly, 4^{255}\equiv 4^3 (\bmod 49)


    So 144^{255}+192^{255} \equiv -3^3-4^3 (\bmod 49) \equiv -27-64 (\bmod 49)\equiv -91\equiv 7 (\bmod 49)

    The remainder is 4
    Last edited by Moo; March 21st 2010 at 06:20 PM. Reason: typo, as stated below
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  4. #4
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    thanks

    jus a small typo from ur side

    4^3=64

    u wrote 67

    so remainder shud be 7
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