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 prime, then f(p) =
(A) P-1
(B) P-2
(C) (P+1)/2
(D) (P-1)/2
(E) 2



If P is prime it's only factors are P and 1. So no number below it will have a common factor with it except 1. Therefore answer should just be P-1.

for eg
if p=2, then f(p) = 1 and 1 does not have a common factor with 2

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f(n) has n-1 positive integers less than n maximum value of f(n) = n-1
Now in the case of a prime number all the positive integers each of which is less than prime number will have no positive factor in common with prime number. Thus f(p) attains its maximum value of p-1 when p is a prime number

Alternatively take a prime 5
4 is not a factor
3 is not a factor
2 is not a factor
1 is not a factor

f(5) is 4 A may be true; B is ruled out; C is ruled out, D is ruled out, E is ruled out

only A can be the answer