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
What you have to realize is that the sum of the digits of a number is the number . We can prove this right now. Any number can be represented in base-10 like this: where . Now realize , so you can use properties of mod to find What is significant here? By repeatedly summing the digits, you are repeatedly looking for the remainder when that sum is divided by 9. Either by using the Euler Theorem and the fact that , or just by finding a pattern, you see that . This means that , which, if calculated properly (not that hard to do), gives you a digit of .
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 |