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# The function f is defined for all the positive integers n by

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

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28 Oct 2010, 08:27
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The function f is defined for all the 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 a prime number then f(p)?

A. p-1
B. p-2
C. (p+1)/2
D. (p-1)/2
E. 2
[Reveal] Spoiler: OA
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28 Oct 2010, 08:31
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anilnandyala wrote:
the function f is defined for all the 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 a 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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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

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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 the positive integers n by [#permalink]

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19 Feb 2015, 04:59
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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

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19 Jan 2016, 13:00
anilnandyala wrote:
The function f is defined for all the 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 a prime number then f(p)?

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

consecutive integers are co-prime, this means that the preceding number is the answer --> P-1; Answer A
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Re: The function f is defined for all the positive integers n by [#permalink]

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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 the positive integers n by [#permalink]

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08 Feb 2016, 21:41
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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Re: The function f is defined for all the positive integers n by [#permalink]

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10 Mar 2017, 11:49
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: The function f is defined for all the positive integers n by   [#permalink] 10 Mar 2017, 11:49
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