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Consecutive and minimum pair puzzle

  1. Feb 23, 2009 #1
    Two consecutive positive decimal integers are such that the sum of the digits of each of them is divisible by 11.

    Determine the minimum possible pair of such numbers.
  2. jcsd
  3. Feb 25, 2009 #2
    Let the 2 numbers be:
    t...x y z 99...9
    t...x y (z+1) 00...0

    Suppose n 9's.
    Then 9*n-1 is divisible by 11.
    So, n=5.
    We know the first number is
    t...x y z 99999
    But t+...+x+y+z = 55-45=10. So, the numbers are
    2899999 and 2900000

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