2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5
Problem
is written on the blackboard. The sum of its digits is calculated. Then the sum of the digits of the result is calculated and so on until we have a single digit. If , what is this digit?
Solution
Lemma: The sum of the digits of are congruent to .
Let represent the sum of the digits of . We can prove that by expressing in the form , where s are digits in the decimal expansion of . By using the fact that for all , we get
Therefore .
Since we repeatedly compute the sum of the digits of , the value of each term will be congruent to . Therefore, we only need to find such that . Using Euler's formula, we know that . Therefore,
So our answer is .
- mako17
See also
2014 UNM-PNM Contest II (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNM-PNM Problems and Solutions |