Northeastern WOOTers Mock AIME I Problems/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 .
We claim that if and only if is prime or .
IF: If is prime, then , which is true for all . For the statement is true as well.
ONLY IF: If is not prime and not , 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 must be , contradiction. It follows that
This proves both directions of the claim. All that remains is to sum the first prime numbers and add to obtain an answer of .