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Math Help - modular arithmetic help

  1. #1
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    modular arithmetic help

    Hey everyone, I am finding these difficult and need some help solving these by hand.

    1. x \equiv 10^{100} \ mod \ 1001, so I want the remainder of \frac{10^{100}}{1001}.

    2. solve 26^{24} \equiv 1 \ (mod \ 35), so I am trying to use Euler's Theorem with a=26 and n=35.


    Any help would be very nice.

    Thanks.
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  2. #2
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    Quote Originally Posted by Nguyen View Post
    Hey everyone, I am finding these difficult and need some help solving these by hand.

    1. x \equiv 10^{100} \ mod \ 1001, so I want the remainder of \frac{10^{100}}{1001}.

    2. solve 26^{24} \equiv 1 \ (mod \ 35), so I am trying to use Euler's Theorem with a=26 and n=35..
    Outline strategy for 1.: 1001 = 7\times 11\times 13, so use Fermat's little theorem to find 10^{100} mod 7, mod 11 and mod 13. Then use the Chinese remainder theorem to piece the results together.

    For 2., you can't "solve" something that isn't an equation. But if you want to verify that 26^{24} \equiv 1 \pmod  {35} then use a similar strategy to 1., namely factorise 35 (and check that the result holds mod 5 and mod 7).
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