Difference between revisions of "1975 IMO Problems/Problem 4"
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Therefore <math>4444^{4444}</math> has less than 17776 digits. This shows that <math>A<9\cdot 17775=159975</math>. The sum of the digits of <math>A</math> is then maximized when <math>A=99999</math>, so <math>B\leq 45</math>. Note that out of all of the positive integers less than or equal to 45, the maximal sum of the digits is 12. | Therefore <math>4444^{4444}</math> has less than 17776 digits. This shows that <math>A<9\cdot 17775=159975</math>. The sum of the digits of <math>A</math> is then maximized when <math>A=99999</math>, so <math>B\leq 45</math>. 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 <math>4444^{4444}\equiv A\equiv B\bmod{9}</math>. This motivates us to compute <math>4444^{4444}\bmod{9}</math>. The easiest way to do this is by searching for a pattern. Note that | + | It's not hard to prove that any base-10 number is equivalent to the sum of its digits modulo 9. Therefore <math>4444^{4444}\equiv A\equiv B(\bmod{9})</math>. This motivates us to compute <math>X</math>, where <math>1\leq X \leq 12</math>, such that <math>4444^{4444}\equiv X(\bmod{9})</math>. The easiest way to do this is by searching for a pattern. Note that |
− | <cmath>4444^1\equiv 7\bmod 9 | + | <cmath>4444^1\equiv 7(\bmod 9)\\4444^2\equiv 4(\bmod 9)\\4444^3\equiv 1(\bmod 9)</cmath> |
and since <math>4444=3\times 1481+1</math>, | and since <math>4444=3\times 1481+1</math>, | ||
− | <cmath>4444^{4444}\equiv 4444^{(3\times1481+1)}\equiv \left(4444^3\right)^{1481}\times 4444\equiv 1\times 4444\equiv 7\bmod{9}</cmath> | + | <cmath>4444^{4444}\equiv 4444^{(3\times1481+1)}\equiv \left(4444^3\right)^{1481}\times 4444\equiv 1\times 4444\equiv 7(\bmod{9})</cmath> |
− | + | Thus, <math>X=7</math>, which means that the sum of the digits of <math>B</math> is <math>\boxed{7}</math>. | |
{{alternate solutions}} | {{alternate solutions}} |
Revision as of 23:15, 22 July 2013
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
Note that
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 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.
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 |