Difference between revisions of "Northeastern WOOTers Mock AIME I Problems/Problem 15"
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− | We claim that <math>\phi(n)>n-\sqrt{n}</math> if and only if <math>n</math> is prime or | + | We claim that <math>\phi(n)>n-\sqrt{n}</math> if and only if <math>n</math> is prime or 1. |
'''IF:''' If <math>n</math> is prime, then <math>\phi(n) = n - 1 > n - \sqrt{n}</math>, which is true for all <math>n \geq 2</math>. For <math>n = 1</math> the statement is true as well. | '''IF:''' If <math>n</math> is prime, then <math>\phi(n) = n - 1 > n - \sqrt{n}</math>, which is true for all <math>n \geq 2</math>. For <math>n = 1</math> the statement is true as well. |
Revision as of 20:33, 7 August 2021
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 or 1.
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 would be , contradiction. It follows that
This proves both directions of the claim. All that remains is to sum the first prime numbers, starting with and ending with , and add to obtain an answer of .