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# M23-35

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Math Expert
Joined: 02 Sep 2009
Posts: 41873

Kudos [?]: 128621 [0], given: 12180

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16 Sep 2014, 01:20
Expert's post
7
This post was
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Difficulty:

45% (medium)

Question Stats:

64% (00:59) correct 36% (01:39) wrong based on 61 sessions

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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. $$\frac{(x + 1)}{2}$$
D. $$\frac{(x - 1)}{2}$$
E. 2
[Reveal] Spoiler: OA

_________________

Kudos [?]: 128621 [0], given: 12180

Math Expert
Joined: 02 Sep 2009
Posts: 41873

Kudos [?]: 128621 [1], given: 12180

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16 Sep 2014, 01:20
1
KUDOS
Expert's post
Official Solution:

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. $$\frac{(x + 1)}{2}$$
D. $$\frac{(x - 1)}{2}$$
E. 2

Basically the question is: how many positive integers are less than given prime number $$x$$, which has no common factor with $$x$$ except 1.

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

For example consider $$x=7=prime$$: how many numbers are less than 7 and have no common factors with 7: 1, 2, 3, 4, 5, and 6, so total of $$7-1=6$$ numbers.

_________________

Kudos [?]: 128621 [1], given: 12180

Current Student
Joined: 06 Jun 2013
Posts: 106

Kudos [?]: 34 [0], given: 72

Concentration: Technology, General Management
GMAT 1: 680 Q48 V35
GPA: 3.47

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15 Jun 2015, 06:33
Hi Bunuel,

Aren't we supposed to return the set of all the positive numbers less than x that does not have a common factor with x OTHER THAN 1.

In that case, aren't we supposed to ignore the positive factor 1 and find out how many other numbers less than x are there who do not have common factor with X.

With this as my understanding, I selected option (A) -> x-2.

eg: say prime number is 7.

Numbers less than 7 other than 1 -> 2, 3,4,5,6.
Total - 5 and hence X-2.

Bunuel wrote:
Official Solution:

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. $$\frac{(x + 1)}{2}$$
D. $$\frac{(x - 1)}{2}$$
E. 2

Basically the question is: how many positive integers are less than given prime number $$x$$, which has no common factor with $$x$$ except 1.

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

For example consider $$x=7=prime$$: how many numbers are less than 7 and have no common factors with 7: 1, 2, 3, 4, 5, and 6, so total of $$7-1=6$$ numbers.

Kudos [?]: 34 [0], given: 72

Math Expert
Joined: 02 Sep 2009
Posts: 41873

Kudos [?]: 128621 [0], given: 12180

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15 Jun 2015, 06:43
svijayaug12 wrote:
Hi Bunuel,

Aren't we supposed to return the set of all the positive numbers less than x that does not have a common factor with x OTHER THAN 1.

In that case, aren't we supposed to ignore the positive factor 1 and find out how many other numbers less than x are there who do not have common factor with X.

With this as my understanding, I selected option (A) -> x-2.

eg: say prime number is 7.

Numbers less than 7 other than 1 -> 2, 3,4,5,6.
Total - 5 and hence X-2.

Bunuel wrote:
Official Solution:

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. $$\frac{(x + 1)}{2}$$
D. $$\frac{(x - 1)}{2}$$
E. 2

Basically the question is: how many positive integers are less than given prime number $$x$$, which has no common factor with $$x$$ except 1.

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

For example consider $$x=7=prime$$: how many numbers are less than 7 and have no common factors with 7: 1, 2, 3, 4, 5, and 6, so total of $$7-1=6$$ numbers.

1 also does not have have a common factor with 7 other than 1. So, you should include 1 as well.
_________________

Kudos [?]: 128621 [0], given: 12180

Manager
Joined: 11 Feb 2015
Posts: 119

Kudos [?]: 32 [0], given: 70

GMAT 1: 710 Q48 V38

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28 Jul 2015, 08:17
I chose A for same reason.
Question doesnt seem very clear whether 1 should be included or not.

Kudos [?]: 32 [0], given: 70

Math Expert
Joined: 02 Sep 2009
Posts: 41873

Kudos [?]: 128621 [0], given: 12180

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28 Jul 2015, 08:26
rukna wrote:
I chose A for same reason.
Question doesnt seem very clear whether 1 should be included or not.

There is no reason whatsoever to exclude 1: 1 also does not have have a common factor with 7 other than 1. So, you should include 1 as well.
_________________

Kudos [?]: 128621 [0], given: 12180

Manager
Joined: 11 Feb 2015
Posts: 119

Kudos [?]: 32 [0], given: 70

GMAT 1: 710 Q48 V38

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31 Jul 2015, 09:27
Bunuel wrote:
rukna wrote:
I chose A for same reason.
Question doesnt seem very clear whether 1 should be included or not.

There is no reason whatsoever to exclude 1: 1 also does not have have a common factor with 7 other than 1. So, you should include 1 as well.

yeah, you are right. Got it.
Thanks Brunel

Kudos [?]: 32 [0], given: 70

Manager
Joined: 05 Jul 2015
Posts: 107

Kudos [?]: 27 [0], given: 3

GMAT 1: 600 Q33 V40
GPA: 3.3

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10 Jan 2016, 17:57
I think this is a poor-quality question and I agree with explanation. This doesn't seem like a 700 level question.

Kudos [?]: 27 [0], given: 3

Intern
Joined: 06 Mar 2016
Posts: 1

Kudos [?]: [0], given: 1

Location: Hong Kong
GMAT 1: 600 Q44 V28
GPA: 2.54

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01 Apr 2016, 04:13
I think this is a poor-quality question and I agree with explanation.

Kudos [?]: [0], given: 1

Senior Manager
Status: One Last Shot !!!
Joined: 04 May 2014
Posts: 254

Kudos [?]: 127 [0], given: 142

Location: India
Concentration: Marketing, Social Entrepreneurship
GMAT 1: 630 Q44 V32
GMAT 2: 680 Q47 V35

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21 Oct 2016, 21:22
I think this is a very good GMAT like 'convoluted' question. The controversy related to 1 is actually because of the confusing language the question stem uses. Let me try and put it in another way:

...and do not have a common factor with $$x$$ other than 1

The above statement only says that 1 is a factor of numbers less than $$x$$ and not that 1 should be excluded from the set of number less than $$x$$

Lets take $$x = 7$$. Below is the set of $$x$$ according to question stem

6 - Less than 7, and does not have a common factor with 7 other than 1
5 - Less than 7, and does not have a common factor with 7 other than 1
4 - Less than 7, and does not have a common factor with 7 other than 1
3 - Less than 7, and does not have a common factor with 7 other than 1
2 - Less than 7, and does not have a common factor with 7 other than 1
1 - Less than 7, and does not have a common factor with 7 other than 1

Hence, 1 has to be in the set!

Total elements: 6
$$x-1$$

Option B
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Kudos [?]: 127 [0], given: 142

Intern
Joined: 05 Jul 2015
Posts: 3

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14 Aug 2017, 05:21
First express the number N in its prime factors in the form:

N = a^p b^q c^r ...

where a, b, c, ... are different prime numbers and p, q, r, ... are
positive integers.

Then the number of positive integers less than N and prime to it is:

phi(N) = N(1 - 1/a)(1 - 1/b)(1 - 1/c).......

This is called Euler's function

Kudos [?]: [0], given: 2

Re: M23-35   [#permalink] 14 Aug 2017, 05:21
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# M23-35

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