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Math Help - remainders over divisions

  1. #1
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    remainders over divisions

    Hey guys, if i had a question of say:

    QUESTION: find the remainders when the following numbers are divided by 89.

    i) 809

    ii) 800^8

    is there an easy was to work this out?

    thanks!
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  2. #2
    MHF Contributor
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    First note that -a\equiv b-a\pmod{b}.

    It is easy to see that 10 * 89 = 809 + 81. Move 81 to the left side and consider both sides modulo 89.

    If you represent 800 as a * 89 + b, then 800^8=(a * 89 + b)^8. By the Binomial theorem (or just by expanding) one sees that all terms have the factor 89 except for the last one, namely, b^8.
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