1975 IMO Problems/Problem 4
Problem
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.)
Solution
Let . We now take the base-10 logarithm of
:
Therefore has less 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 provethat any base-10 number is equivalent to the sum of its digits modulo 9. Therefore . We now compute
:
After expanding, every term except is divisible by 9, so they all cancel out. This shows that
. Note that
. Therefore
After expanding, every term except is divisible by 9, so they all cancel out. This shows that
. Therefore
, and the sum of the digits of
is also
. However, we established that the sum of the digits of
is at most 12. This proves 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.
See Also
1975 IMO (Problems) • Resources | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All IMO Problems and Solutions |