Northeastern WOOTers Mock AIME I Problems/Problem 15
Problem 15
Find the sum of all integers such that where denotes the number of integers less than or equal to that are relatively prime to .
Solution
We claim that if and only if is prime. \
IF: If is prime, then , which is true for all . \
ONLY IF: If is not prime, then must have a prime divisor such that ; if this was not the case, then the number of not necessarily distinct prime factors could have would be , contradiction. It follows that .