Consecutive and minimum pair puzzle

[K Sengupta]

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.

 

[Rogerio]

Let the 2 numbers be:
t...x y z 99...9
and
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