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Math Help - natural number divisible by 2004

  1. #1
    Member disclaimer's Avatar
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    natural number divisible by 2004

    Hi all;

    Prove that there's a natural number divisible by 2004 consisting only of zeros and sevens.

    Help appreciated. Thank you.
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  2. #2
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    Quote Originally Posted by disclaimer View Post
    Hi all;

    Prove that there's a natural number divisible by 2004 consisting only of zeros and sevens.
    Let a_1 = 7, a_2 = 77, a_3 = 777, .... a_{2005} = 77 ... 7.

    By pigeonhole principle there are two numbers with same remainder. Thus there are i>j with a_i - a_j divisible by 2004. And this number only consists of only sevens and zeros.
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