Like all function/wacky symbol questions, the key is understanding the definition.
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.
So, we count the number of positive integers less than n which have NO factors in common with n (other than 1).
For example:
f(6)
The numbers below 6 are 1, 2, 3, 4 and 5. The factors of 6 are {1, 2, 3, 6}. 2, 3 and 4 all have factors in common with 6, leaving 1 and 5 with no factors (other than 1) in common with 6.
Therefore, f(6) = 2
For most numbers, calculating f(n) will be painful. However the question asks about f(p), where p is a prime number.
The first thing to note is that the answer choices contain variables (except (e), which we should eliminate pretty quickly using common sense).
So, one way we could attack this question is to pick numbers.
Let's try p=5
So, the numbers smaller than p are {1, 2, 3, 4}. Since 5 is prime, none of those will have factors in common with p. So, f(5)=4.
Subbing p=5 into the choices:
(a) p-1 = 4
(b) p-2 = 3
(c) (p+1)/2 = 3
(d) (p-1)/2 = 2
(e) 2
Only (a) is a match: choose (a).
If more than one choice matched, we would have tried another p value to narrow it down.
Of course, if you see the logical solution, that's quicker than picking numbers. We know that, by definition, primes have exactly 2 factors: 1 and the prime itself. So, every prime will have no factors in common (other than 1) with any numbers smaller than the prime.
Since there are (p-1) positive integers smaller than p, f(p) = p-1: choose (a).
