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30 Dec 2009, 08:05
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If \(x\) is a positive integer, \(f(x)\) is defined as the number of positive integers smaller than \(x\) that share no common factors with \(x\) other than 1. If \(x\) is a prime number, what is the value of \(f(x)\) in terms of \(x\)?
A. \(x - 2\)
B. \(x - 1\)
C. \(\frac{(x + 1)}{2}\)
D. \(\frac{(x - 1)}{2}\)
E. 2
M23-35
A. \(x - 2\)
B. \(x - 1\)
C. \(\frac{(x + 1)}{2}\)
D. \(\frac{(x - 1)}{2}\)
E. 2
M23-35
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30 Dec 2009, 08:12
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If \(x\) is a positive integer, \(f(x)\) is defined as the number of positive integers smaller than \(x\) that share no common factors with \(x\) other than 1. If \(x\) is a prime number, what is the value of \(f(x)\) in terms of \(x\)?
A. \(x - 2\)
B. \(x - 1\)
C. \(\frac{(x + 1)}{2}\)
D. \(\frac{(x - 1)}{2}\)
E. 2
The question is essentially asking how many positive integers smaller than a given prime number \(x\) have no factors in common with \(x\), except for 1.
Since \(x\) is a prime number, it only has two distinct factors: 1 and itself. As a result, any positive integer smaller than \(x\) will not share any factors with \(x\) other than 1. Therefore, there will be \(x-1\) positive integers smaller than \(x\) that meet this criterion.
For example, consider the prime number \(x=7\). The positive integers smaller than 7 that have no factors in common with 7, except for 1, are: 1, 2, 3, 4, 5, and 6. Thus, the value of \(f(x)\) for \(x=7\) is \(7-1=6\).
Answer: B
General Discussion
06 Jan 2017, 04:34
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sagarsabnis
We are looking for the number of co-prime factors of x which are less than x. Note that if x is prime, all positive integers less than it will be co-prime to it. That is because this is what a prime number is - not divisible by any number smaller than it except 1.
So all positive integers less than x (including 1) will have no common factor with x other than 1. There will be (x - 1) such positive integers.
Answer (B)
