If x is a positive integer, f(x) is defined as the number of positive

Question

If x is a positive integer, f(x) is defined as the number of positive integers which are less than x and do not have a common factor with x other than 1. If x is prime, then f(x) =?

A. x-2
B. x-1
C. (x+1)/2
D. (x-1)/2
E. 2

A.  x - 2  
B.  x - 1  
C.  \frac{(x + 1)}{2}  
D.  \frac{(x - 1)}{2}  
E. 2

M23-35

Bunuel · Accepted Answer

If x is a positive integer, f(x) is defined as the number of positive integers which are less than x and do not have a common factor with x other than 1. If x is prime, then f(x) =?

A. x-2
B. x-1
C. (x+1)/2
D. (x-1)/2
E. 2

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

KarishmaB · Answer

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)

ScottTargetTestPrep · Answer

We are given that f(x) is defined as the number of positive integers that are less than x and do not have a common factor with x other than 1. We also are given that x is a prime number and we must determine the value of f(x).

Since x can be any prime number, let’s let x = 5. The positive integers less than 5 are 1, 2, 3, and 4. Note that those integers do not have a common factor with 5, other than 1. Thus, f(x) = 4.

Plugging 5 in for x in our answer choices, we see that the only answer choice that equals 4 is B, x - 1.

Answer: B

MS61 · Answer

How x-1 true in case of prime number 2?

Between 2 & 1 there are zero positive numbers and 2-1 = 1

MS61 · Answer

How x-1 true in case of prime number 2?

Between 2 & 1 there are zero positive numbers and 2-1 = 1

Bunuel · Answer

If x = 2, the number of positive integers smaller than 2 that share no common factors with 2 other than 1 is one, and that integer is 1, which does not share any common factor with 2 except 1.

Samriddhi0603 · Answer

why is number 1 counted since it is explicitly stated that it dosen't share any other factor except 1, it should be excluded.

Bunuel · Answer

You're misunderstanding the phrase "other than 1" in the question. It doesn’t mean to exclude 1 from the count. It simply clarifies that the only common factor these numbers have with x is 1. If x is 7, then the number of positive integers smaller than x that share no common factors with x other than 1 is six: 1, 2, 3, 4, 5, and 6.

arunbhati · Answer

Dear Sir,

(X-1)/2 is wrong only when we use X=2

Bunuel · Answer

No. 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 \(7-1=6\).

While (x - 1)/2 gives 3.

While (x - 1)/2 gives 3.

arunbhati · Answer

Hi Sir

3 and 7 doesn’t have any common factor except 1

Bunuel · Answer

I’m not sure what you mean. You said "(X-1)/2 is wrong only when we use X=2", but I showed with x = 7 that (x - 1)/2 does not give the correct value of 6. Please review the question again.

arunbhati · Answer

Hi Sir,

Please confirm my understanding relates to questions is correct or not

X is prime number and f(x) should be less than X and whose value doesn’t share common factor except 1

Here X=7 and F(x)=(7-1)/2=3

7 and 3 don’t have any common factor except 1

Where I am wrong with option (X-1)/2

Bunuel · Answer

It seems you ignored the discussion above and have not studied the solutions provided carefully enough.

f(x) is  not  asking for one number relatively prime to x.

f(x) is the  number  of all positive integers less than x that are coprime with x.

If x is a positive integer, f(x) is defined as the number of positive

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