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Math Help - modulo

  1. #1
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    modulo

    find the remainder when 11121 is divided by 13
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  2. #2
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    Re: modulo

    use 11 = -2 mod 13 and 2^5 = -1 mod 13.
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  3. #3
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    Re: modulo

    Hello, franios!

    \text{Find the remainder when }11^{121}\text{ is divided by 13.}

    Note that: . 11 \:\equiv\: \text{-}2\text{ (mod 13)}

    Also that: . 11^6 \:=\:(\text{-}2)^6\text{ (mod 13)} \:=\:64 \quad\Rightarrow\quad 11^6 \:\equiv\: \text{-}1\text{ (mod 13)}

    Hence: . 11^{121} \:=\:11^{6\cdot20+1} \;=\;11^{6\cdot20}\cdot 11^1 \;=\;(11^6)^{20}\cdot 11

    Therefore: . 11^{121} \;\equiv\;(\text{-}1)^{20}\cdot 11\text{ (mod 13)} \;\equiv\;11\text{ (mod 13)}
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