Problem Solving

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 a prime number then f(p)=
A. p-1
B. p-2
C. (p+1)/2
D. (p-1)/2
E. 2

If not the wording the question wouldn't be as tough as it is now. The GMAT often hides some simple concept in complicated way of delivering it.

This question for instance basically asks: how many positive integers are less than given prime number p which have no common factor with p except 1.

Well as p is a prime, all positive numbers less than p have no common factors with p (except common factor 1). So there would be p-1 such numbers (as we are looking number of integers less than p).

For example: if p=7 how many numbers are less than 7 having no common factors with 7: 1, 2, 3, 4, 5, 6 --> 7-1=6.

Answer: A

Now the options are:

(A) P-1
(B) P-2
(C) (P+1)/2
(D) (P-1)/2
(E) 2

My question is why cant it be B isn't that all numbers less than P and since P is a prime number they don't have common factor as well!