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Math Help - repunits

  1. #1
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    Exclamation repunits

    Find the smallest multiple of 17 that has only 1s in its decimal expansion.
    Hint: the numbers in question are of the form (10^n-1)/9.

    How should I solve this question? Please teach me . Thank you.
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  2. #2
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    Hello, Jenny!

    Find the smallest multiple of 17 that has only 1's in its decimal expansion.
    Hint: the numbers in question are of the form (10^n - 1)/9

    I haven't found a way to use that hint.
    If you're desperate, here's a very primitive method.

    Divide 17 into a "string of 1's" ... and wait for it to come out even.
    Code:
                    6 5 3 5 9 4 7 7 1 2 4 1 8 3
              ---------------------------------
          1 7 ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
               --------------------------------
                                              0

    Therefore: 1,111,111,111,111,111 is the smallest multiple of 17.

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  3. #3
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    Hi Soroban,
    Thank you very much. I hope that I can use the hint to solve this question.

    My friend came up this :

    17n = (10^n-1)/9

    Is it useful regarding to the hint?
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  4. #4
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    Quote Originally Posted by Jenny20 View Post
    Hi Soroban,
    Thank you very much. I hope that I can use the hint to solve this question.

    My friend came up this :

    17n = (10^n-1)/9

    Is it useful regarding to the hint?
    I think you mean,
    17m=(10^n-1)/9

    The number on left is multiple of 17.
    The number on right is a repunit.

    Multiply by 9,
    153m=10^n-1
    Thus,
    153m+1=10^n

    Note, the expression on right is 1 followed by zeros.
    Thus, find the smallest "m" that will make,
    153m+1
    Be followed by zeros.
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  5. #5
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    Hi perfecthacker,

    Can you use the " hint " to show me the way to solve my question? If possible, please do show me the working steps. Thank you very much.
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  6. #6
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    Quote Originally Posted by Jenny20 View Post
    Hi perfecthacker,

    Can you use the " hint " to show me the way to solve my question? If possible, please do show me the working steps. Thank you very much.
    You can combine the hint with soroban's method, as:

    17.N=(10^n-1)/9

    means:

    153.N=10^n-1,

    but the rigt hand side is a number all of whoes digits are 9, so you can do
    the long division with 153 instead of 17 and a string of 9's instead of 1's
    in soroban's method

    RonL
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