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, Answer Key) | ||
Preceded by: Problem 33 |
Followed by: Problem 35 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • U1 • U2 • U3 • U4 • U5 • U6 • U7 • U8 • U9 • U10 |