The function f is defined for all positive integers n

[Harish Dorai]

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 also 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

[givemeanid]

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 also 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

Take a prime number and figure out a specific soln for that prime number.
Let p = 5. So, excluding 1, the other numbers that have no factors common with 5 are 2,3,4.
Let p = 7. So, excluding 1, the other numbers that have no factors common with 7 are 2,3,4,5,6
Do you see the pattern? For any prime number, all the numbers less than it will have no factors in common with it except 1.
So f(p) = p - 2
Answer is B.

[guest]

Slightly confused...isn't 1 also included since it qualifies as having no common factor other than 1?

Please explain.

[JAMGAJR]

In my opinion p-1 is the answer

you must consider 1............

[Harish Dorai]

I also came up with the answer p-2, but as JAMGAJR mentioned, we need to include 1 and hence the correct answer is p-1 (A).

[givemeanid]

Slightly confused...isn't 1 also included since it qualifies as having no common factor other than 1?

Please explain.

#$@@#!#%^. Careless mistake again. You are right. 1 should be included.

[thompson]

1. the ans of OA is "A".

2. 1 is not prime number.

3. n and f(n) can't have any factor in common. if n is a prime integer and f(n)=n-1, the GCD of n and of (n-1) is 1. >> my idea (what is the term of this situation ?)

sorry, I'm not a natural speaker in English.
if i make a mistake, please keep me informed .

[StaceyKoprince]

You are right that 1 is not a prime number - that's not what the others meant by needing to include 1. This is what they meant:

Pick a prime number for p. Let's say p=5.

The positive integers less than 5 are 4, 3, 2, and 1.

5 and 4 share only 1 as a factor
5 and 3 share only 1 as a factor
5 and 2 share only 1 as a factor
5 and 1 share only 1 as a factor

There are four positive integers, therefore, that are both less than 5 and share only 1 as a factor. In other words, we include 1 in this set of integers.

So, yes, as you said, the answer is A. The second poster, up above, made a mistake by forgetting to include 1 in the set and the others were just correcting the mistake.