2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5
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?
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
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 .
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