1975 IMO Problems/Problem 4
When is written in decimal notation, the sum of its digits is . Let be the sum of the digits of . Find the sum of the digits of . ( and are written in decimal notation.)
Therefore has fewer than 17776 digits. This shows that . The sum of the digits of is then maximized when , so . Note that out of all of the positive integers less than or equal to 45, the maximal sum of the digits is 12.
It's not hard to prove that any base-10 number is equivalent to the sum of its digits modulo 9. Therefore . This motivates us to compute , where , such that . The easiest way to do this is by searching for a pattern. Note that
and since ,
Thus, , which means that the sum of the digits of is .
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
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