Let s be the smallest positive integer with the property that its digit sum and the digit sum of s + 1 are both divisible by 19. How many digits does s have?
Our number s is composed of a left portion and a right portion.
The right portion is a string of 9s, and call the number of 9s "k".
The left portion is an integer which does not end in 9.
The sod(s) - sod(s-1) = 9k - 1 which is 0 mod 19
The smallest k that works is 17.
A string of 17 9s has an sod of 153 which is 1 mod 19.
The left portion must have an sod of 18 (and not end in 9), the smallest such integer is 198.
s = 19899999999999999999
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Posted by Larry
on 2024-09-15 15:57:42 |
Solution
