Consecutive and minimum pair puzzle

AI Thread Summary
Two consecutive positive decimal integers can be found where the sum of the digits of each is divisible by 11. The minimum possible pair identified is 2899999 and 2900000. The analysis shows that for a number with n nines, the equation 9*n-1 must be divisible by 11, leading to n=5. The digit sum condition is satisfied with t+x+y+z equaling 10. This puzzle illustrates a unique relationship between digit sums and divisibility rules.
K Sengupta
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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.
 
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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

:smile:
 

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