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common with n other than 1

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by sanju09 » Fri Mar 18, 2011 3:05 am
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



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Source: — Problem Solving |

by vineeshp » Fri Mar 18, 2011 3:28 am
I go for A, p-1.

If p is a prime number, none of the numbers from 1 to p-1 have a common factor with p other than 1. So the number of such numbers is p-1.
Vineesh,
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by Anurag@Gurome » Fri Mar 18, 2011 6:02 am
Let us take any prime number, say, p = 7
Then positive integers less than 7 are 1, 2, 3, 4, 5, 6.

7 and 1 have 1 as a common factor.
7 and 2 have 1 as a common factor.
7 and 3 have 1 as a common factor.
7 and 4 have 1 as a common factor.
7 and 5 have 1 as a common factor.
7 and 6 have 1 as a common factor.
So, it can be seen that if p is a prime number then it's factor are only p and 1. So, there will be no number less than p, which will have a common factor with it except 1. So, f(p) = p - 1

The correct answer is A.
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