9 thoughts on “Applying Negative Remainders

  1. Sanchi, rather than solving I will give you a hint. Take 12 common in numerator and denominator so you are left with (2 * 24^59 + 5 * 60^23) / 7. This should be easy to solve – find the answer and multiply by 12.

    regards
    J

    • Answer should be 24, right?

      24^60 = 12^60 * 2^60 div by 12*7 => 12 * Rem (2^60/7) => rem = 12*1 = 12

      60^ 24 = 12^60 * 5^60 div by 12*7 => 12* Rem (5^60/7) => rem = 12* (-1)^20 = 12

      Hence, (12+12) div by 84 gives the total remainder => Remainder = 24

      • Not quite, you’ve made a slight error!
        You wrote:
        24^60 = 12^60 * 2^60 div by 12*7 => 12 * Rem (2^60/7) => rem = 12*1 = 12
        But actually
        24^60 = 12^60 * 2^60 div by 12*7 => 12 * Rem [(12^59 * 2^60)/7] => rem = 12*3 = 36
        Similarly the other also will give 36 and hence the overall remainder will turn out to be 72.

        regards
        J

    • So if you are dividing by 13, just start the normal procedure and keep going till you reach either 1 or 2. If you reach one, your job is done, if you reach 12, treat it as -1 and continue (otherwise you would have to go twice as far to reach 1). Deal on a case-by-case basis!

      regards
      J

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