Adding 0 makes no change, Really ?

Number Theory Level 3

Given that in the 8-digit number \(\overline{\mathrm{ABCDEFGH}}\),

(i) \(\overline{\mathrm{ABCD}}=\overline{\mathrm{EFGH}}\)
(ii) The numbers \(\overline{\mathrm{ABCDEFGH}}\) and \(\overline{\mathrm{ABCD0EFGH}}\) are both divisible by \(11\).
Let the sum of all possible values of \(\overline{\mathrm{ABCDEFGH}}\) be \(N\).

Find the digit sum of \(N\).

Details and assumptions:-

  • \(\overline{\mathrm{ABC}}\) means the number in decimal representation with digits \(A,B,C\) i.e. \(\overline{\mathrm{ABC}} = \mathrm{100A+10B+C}\)
  • The letters \(A\) to \(H\) do not necessarily stand for distinct digits.
  • In the second number, the digit \(0\) is added in the middle of the 8-digit number, which makes it a 9-digit number.
  • Digit sum is sum of all digits in decimal representation, digit sum of \(12023\) is \(1+2+0+2+3=8\)
  • 00123 is not a 5-digit number.

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