# 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.