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Math Help - modular arith

  1. #1
    Senior Member sfspitfire23's Avatar
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    modular arith

    Find the remainder of 100^{11}+1 divided by 11

    100 is equivalent to 1 (mod 11) and 1 is equivalent to -10 (mod 11).

    Thus, 1^{11}+-10=-9. But -9 is equivalent to 2 (mod 11). Thus, the remainder is 2.

    Correct?
    Last edited by sfspitfire23; August 3rd 2010 at 11:14 AM. Reason: spelling
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  2. #2
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by sfspitfire23 View Post
    Find the remainder of 100^{11}+1 divided by 11

    100 is equivalent to 1 (mod 11) and 1 is equivalent to -10 (mod 11).

    Thus, 1^{11}+-10=-9. But -9 is equivalent to 2 (mod 11). Thus, the remainder is 2.

    Correct?
    Why convert the 1 to -10? You create extra work for yourself. Just keep it as 1 + 1 = 2
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  3. #3
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    Quote Originally Posted by sfspitfire23 View Post
    Find the remainder of 100^{11}+1 divided by 11

    100 is equivalent to 1 (mod 11) and 1 is equivalent to -10 (mod 11).

    Thus, 1^{11}+-10=-9. But -9 is equivalent to 2 (mod 11). Thus, the remainder is 2.

    Correct?
    For similar cases you could also use Fermat's Little Theorem. Since 100 and 11 are relatively prime with 11 a prime, we have 100^{10}\equiv 1(mod\ 11) , then 100^{11}+1\equiv 100+1(mod\ 11) and 100 is congruent to 1 modulo 11. So 100^{11}+1\equiv 2 (mod\ 11).
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