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Intern  Joined: 07 Dec 2009
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The function F is defined for all positive integers n by the following  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 63% (01:42) correct 37% (01:51) wrong based on 698 sessions

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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.
Edited the question
Math Expert V
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Re: The function F is defined for all positive integers n by the following  [#permalink]

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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.

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

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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.

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Manager  Joined: 01 Nov 2010
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Re: The function F is defined for all positive integers n by the following  [#permalink]

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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

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

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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.

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.

Hope it's clear.
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GMAT 1: 700 Q48 V37 GMAT 2: 720 Q48 V40 Re: The function F is defined for all positive integers n by the following  [#permalink]

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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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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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5
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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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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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

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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: The function F is defined for all positive integers n by the following  [#permalink]

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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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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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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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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.

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