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Math Help - remainder

  1. #1
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    remainder

    Find the remainder when 111203 is divided by 13
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  2. #2
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    Re: remainder

    Hey franios.

    You can do this quickly on a computer but do you have to do this using specific techniques?
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  3. #3
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    Re: remainder

    yes i believe we have to do it using mods
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  4. #4
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    Re: remainder

    Do you know how to do long division with regular numbers?
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  5. #5
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    Re: remainder

    this is how i do such things: i get a calculator and check 111203/13 = 8554.0769230769230769230769230769

    i don't care about the decimal part, i only want the integer part. this tells me that:

    111203 = (111203) - (8554)(13) (mod 13).

    now (8554)(13) = 111202, and 111203 - 11202 = 1, so:

    11203 = 1 (mod 13)

    if you don't have access to a calculator while doing this, you can do it this way:

    111203 < 13000 = 13*10000 and

    111203 > 1300 = 13*1000.

    so let's find the largest "n" with 13*n*1000 < 111203.

    we guess 5 first: 13*5000 = 65000, so n is at least 5
    we guess 7 or 8, next: let's guess 7: 13*7000 = 91000, so n is at least 7
    let's say we not sure if it's 7,8,or 9, and we guess 8: 13*8000 = 104000, so n is at least 8.

    now we know n is 8 or 9, let's check 9: 13*9000 = 117000, so 9 is too big. thus n = 8.

    so 111203 = 111203 - 104000 (mod 13) that is:
    11203 = 7203 (mod 13) <--7203 is a LOT smaller than 11203.

    we do the same thing as before, except now we're looking for 13*n*100 < 7203.

    6500 < 7203, so n is at least 5.
    9100 > 7203, so n is at most 6.

    7800 > 7203, so n = 5.

    so 111203 = 7203 = 7203 - 6500 = 703 (mod 13) <---703 is again a LOT smaller than 7203.

    again, looking for an n with 13*n*10 < 703

    650 < 703
    780 > 703, so n = 5.

    so 111203 = 7203 = 703 = 703 - 650 = 53 (mod 13) <--again, this is getting nice and small.

    now by inspection, we see: 13*4 = 52 < 53 < 65 = 13*5, so:

    53 = 53 - 52 = 1 (mod 13), and since 0 ≤ 1 < 13, we are done.

    this is just like long division, but we only "keep the remainders" (once we find the n's, we discard them, because we don't need them anymore).
    Last edited by Deveno; December 12th 2012 at 03:55 PM. Reason: mis-read question
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