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The function F is defined for all positive integers n by the following

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The function F is defined for all positive integers n by the following  [#permalink]

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New post Updated on: 23 Apr 2019, 14:49
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The function F is defined for all positive integers n by the following rule: f(n) is the number of positive integer 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

Originally posted by bhavinnc on 24 Jan 2010, 19:38.
Last edited by Bunuel on 23 Apr 2019, 14:49, edited 2 times in total.
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Re: The function F is defined for all positive integers n by the following  [#permalink]

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New post 28 Oct 2010, 08:31
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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

The confusing moment in this question is its wording. Basically question is: how many positive integers are less than given prime number p which has no common factor with p except 1.

Well as p is a prime, all positive numbers less than p have no common factors with p (except common factor 1). So there would be p-1 such numbers (as we are looking number of integers less than p).

If we consider p=7 how many numbers are less than 7 having no common factors with 7: 1, 2, 3, 4, 5, 6 --> 7-1=6.

Answer: A.
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Re: The function F is defined for all positive integers n by the following  [#permalink]

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New post 29 Jan 2012, 16:53
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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

Guys - does this questions makes sense to anyone? I am struggling. Does it mean that:

F(n) is a list of positive integers. AM I right?

for e.g f(5) = 3,4.

I am stuck after this. Can someone please help?
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Re: The function F is defined for all positive integers n by the following  [#permalink]

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New post 09 Nov 2010, 07:46
The moment you put a prime number in the function f(n), notice that all the numbers lesser than n have no divisor clashing with divisor of n since n is prime!!.

For instance f(7)= {6, 5, 4, 3, 2, 1}

Thus for f(p) number of integers falling under this set will be p-1


Answer :- A
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Re: The function F is defined for all positive integers n by the following  [#permalink]

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New post 29 Jan 2012, 16:59
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enigma123 wrote:
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

Guys - does this questions makes sense to anyone? I am struggling. Does it mean that:

F(n) is a list of positive integers. AM I right?

for e.g f(5) = 3,4.

I am stuck after this. Can someone please help?


If not the wording the question wouldn't be as tough as it is now. The GMAT often hides some simple concept in complicated way of delivering it.

This question for instance basically asks: how many positive integers are less than given prime number p which have no common factor with p except 1.

Well as p is a prime, all positive numbers less than p have no common factors with p (except common factor 1). So there would be p-1 such numbers (as we are looking number of integers less than p).

For example: if p=7 how many numbers are less than 7 having no common factors with 7: 1, 2, 3, 4, 5, 6 --> 7-1=6.

Answer: A.

Hope it's clear.
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Re: The function F is defined for all positive integers n by the following  [#permalink]

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New post 04 Apr 2012, 10:14
But it says no other factors in common with n other than 1, why do we have to include 1 then? I thought since 1 is a factor of 1 itself and p, we cannot include it.
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Re: The function F is defined for all positive integers n by the following  [#permalink]

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New post 04 Apr 2012, 10:24
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BN1989 wrote:
But it says no other factors in common with n other than 1, why do we have to include 1 then? I thought since 1 is a factor of 1 itself and p, we cannot include it.


Each positive integer should have no factor common with n except 1.
1 also has only a single factor i.e. 1 common with p. So we do include 1.
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Re: The function F is defined for all positive integers n by the following  [#permalink]

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New post 09 Oct 2013, 09:36
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Okay let's see what happens here.

It took me 20 seconds to understand what the question was precisely asking for. But this is the most important step; do not attempt anything if you don't understand throughly the question.

What I figured is that the definition of "relatively prime" was pretty close to the description assigned to the question.

Two different numbers are said to be relatively prime whenever their GCF=1.

Let's pick a random example: how many numbers less than 105 are relatively prime to 105?

105=3(5)7 then the total number of relatively primes will be: 105(1-1/3)(1-1/5)(1-1/7)=48

Let's apply the same logic to our question and consider a random prime number: p(1-1/p)= p-1 which turns out to be the correct answer.
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Re: The function F is defined for all positive integers n by the following  [#permalink]

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New post 08 Feb 2016, 19:14
well damn...the wording is indeed confusing..as I was thinking that f(n) is the sum of the all numbers..
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Re: The function F is defined for all positive integers n by the following  [#permalink]

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New post 08 Feb 2016, 21:41
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mvictor wrote:
well damn...the wording is indeed confusing..as I was thinking that f(n) is the sum of the all numbers..



In such questions, it is advisable to take an example to figure out what the question is saying.

"The function f is defined for all the positive integers n by the following rule:"

We are looking at all positive integers so say n is 3.


"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 . "

Positive integers less than n -> 1, 2
Both do not have a factor in common with 3.
So f(n) = 2 (number of integers which have nothing in common with n except 1)

"if p is a prime number then f(p)?"

p must be a prime number. Our previous example was a prime number. Let's take another say 5.
Positive integers less than 5 -> 1, 2,3, 4
All 4 integers will have no factor in common with 5 because 5 is prime.
f(5) = 4

This will be the case with all prime numbers. Since a prime has no factor in common (except 1) with all positive integers less than it,
f(p) = p - 1

Answer (A)
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Re: The function F is defined for all positive integers n by the following  [#permalink]

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New post 29 Mar 2018, 12:38
Hi All,

This question is meant to ask: "If P is a prime number, then f(P)= ?" This question can be solved by TESTing VALUES.

Let's TEST N=7. The f(7) = all the positive integers less than 7 that have no factor in common with 7 except for 1.

THAT list is 1, 2, 3, 4, 5, 6 = 6 terms.

Thus, we're looking for an answer that equals 6 when we plug N=7 into it. There's only one answer that matches...

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Re: The function F is defined for all positive integers n by the following  [#permalink]

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New post 26 Feb 2019, 05:37
are there any other questions that talk about position integers?
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Re: The function F is defined for all positive integers n by the following  [#permalink]

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New post 11 Oct 2019, 03:45
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bhavinnc wrote:
The function F is defined for all positive integers n by the following rule: f(n) is the number of positive integer 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


Since p can be ANY PRIME NUMBER, let p=2.
In this case:
f(p) = f(2) = the number of positive integers less than 2 that have no factor in common with 2.
Since only ONE positive integer is less than 2, f(2) = 1.
The correct answer must yield 1 when p=2.
Only A works:
p-1 = 2-1 = 1


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Re: The function F is defined for all positive integers n by the following  [#permalink]

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New post 27 Nov 2019, 12:11
bhavinnc wrote:
The function F is defined for all positive integers n by the following rule: f(n) is the number of positive integer 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



We can see that, in other words, f(n) is the number of positive integers less than n that are relatively prime to n. If p is a prime, then any positive integer less than p will be relatively prime to p. For example, if p = 7, then f(7) = 6 since 1, 2, 3, 4, 5, and 6 are all relatively prime to 7. Therefore, f(p) = p - 1.

Answer: A
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Re: The function F is defined for all positive integers n by the following   [#permalink] 27 Nov 2019, 12:11
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