mysseo wrote:The function f is defined for all positive integers N by the following rule: f(N) is the number of positive integers each of which is less than N and has no positive factor in common with N other than 1. If P is any prime number then f(P) =
A. P-1
B. P-2
C. (P+1)/2
D. (P-1)/2
E. 2
Tricky solution:
Let us take p = 2 (smallest prime)
Now number of positive integers less than p and has no common factor with p other than 1 is 1. So f(2) = 1
Only option
A satisfies this result.
Mathematical Approach:
Note that a prime number will have common factors other than 1 only with its multiples like p², p³ etc. As p is always greater than 1, all multiples of p are greater than p. Hence, none of the integers less than p will have any common factor with p.
Thus, f(p) = Number of positive integers less than p = (p - 1)
The correct answer is
A.
