Two consecutive numbers both have a digital sum which is a multiple of 11. What is the smallest pair of numbers?

Note: the digital sum is the number equalling the sum of all the digits of the number. For example the digital sum of 123456 is 21.

How about 2,899,999 (digit sum 55) and 2,900,000 (digit sum 11)?

The number clearly needs some number of 9's at the end.  If it doesn't end in 9, then the digital sums only differ by 1.  5 but not 6 9's seem right, because 99999 has a digital sum of 45 and 100000 has a digital sum of 1, and 45 - 1 is a multiple of 11.  2,899,999 is the smallest number ending in exactly 5 9's that has a digital number which is a multiple of 11.

Posted by Steve Herman on 2008-12-08 13:28:13