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Math Help - elementary number theory: sum of digits

  1. #1
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    elementary number theory: sum of digits

    let x be an integer and let j be the sum of the digits of x. Subtract j from x and get x-j=y. Let k be the sum of the digits of y. Show that y-77,778j cannot equal 7,777,777.

    I have tried to do this with divisibility criteria. For instance 77,778j must be even and 77,778 is divisible by 3 while 7,777,777 is not but the arbitrariness of x and y leave me stuck.
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  2. #2
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    Actually, y must be divisible by three by the following theorem: Let x be an integer and j be the sum of its digits in base 10. Then x=j (\bmod 3) no matter what. Therefore 3|y=x-j, so 3|y-77778j and 7777777 does not.

    Pf: Let x be an integer. Expressed in base 10, x=a_0+a_110^1+a_210^2+...+a_n10^n for some unique set of integers a_i\in\mathbb{Z}_{10}. Since 10=1 (\bmod 3), 10^k=1 (\bmod 3) for all k, so x=a_0+a_110^1+a_210^2+...+a_n10^n= a_0+a_1(1)+a_2(1)+...+a_n(1)=a_0+a_1+a_2+...+a_n (\bmod 3). QED
    Last edited by Media_Man; October 17th 2009 at 11:42 AM. Reason: incorrect wording; thanks Bingk
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  3. #3
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    Media_Man, you said that y must be divisible by 3, in other words, 3 divides y, which is correct ... but later on, you write that y divides 3
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