2006 iTest Problems/Problem 34
For each positive integer let denote the set of positive integers such that is divisible by . Define the function by the rule Let be the least upper bound of and let be the number of integers such that and . Compute the value of .
Solution
We find that the prime factorization of is .
Then we can compute (where is the Carmichael function) by Carmichael's theorem: it is .
As for solving , we must have odd (otherwise it would not be coprime to ), and we must also have be a primitive root modulo as well as a primitive root modulo . There are primitive roots modulo (where is the Euler totient function) and primitive roots modulo . Then we have by the Chinese Remainder Theorem, so our answer is and we are done.
See also
2006 iTest (Problems) | ||
Preceded by: Problem 33 |
Followed by: Problem 35 | |
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