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Math Expert V
Joined: 02 Sep 2009
Posts: 58445

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4
31 00:00

Difficulty:   95% (hard)

Question Stats: 34% (02:39) correct 66% (02:45) wrong based on 366 sessions

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The functions $$f$$ and $$g$$ are defined for all the positive integers $$n$$ by the following rule: $$f(n)$$ is the number of positive perfect squares less than $$n$$ and $$g(n)$$ is the number of prime numbers less than $$n$$. If $$f(x) + g(x) = 16$$, then $$x$$ is in the range:

A. $$30 \lt x \lt 36$$
B. $$30 \lt x \lt 37$$
C. $$31 \lt x \lt 37$$
D. $$31 \lt x \lt 38$$
E. $$32 \lt x \lt 38$$

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Math Expert V
Joined: 02 Sep 2009
Posts: 58445

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Official Solution:

The functions $$f$$ and $$g$$ are defined for all the positive integers $$n$$ by the following rule: $$f(n)$$ is the number of positive perfect squares less than $$n$$ and $$g(n)$$ is the number of prime numbers less than $$n$$. If $$f(x) + g(x) = 16$$, then $$x$$ is in the range:

A. $$30 \lt x \lt 36$$
B. $$30 \lt x \lt 37$$
C. $$31 \lt x \lt 37$$
D. $$31 \lt x \lt 38$$
E. $$32 \lt x \lt 38$$

Positive perfect squares: 1, 4, 9, 16, 25, 36, ..,

Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ...

If $$x = 31$$, then $$f(31) = 5$$ and $$g(31) = 10$$: $$f(x) + g(x) = 5 + 10 = 15$$.

If $$x = 32$$, then $$f(32) = 5$$ and $$g(32) = 11$$: $$f(x) + g(x) = 5 + 11 = 16$$.

...

If $$x = 36$$, then $$f(36) = 5$$ and $$g(36) = 11$$: $$f(x) + g(x) = 5 + 11 = 16$$.

If $$x = 37$$, then $$f(37) = 6$$ and $$g(37) = 11$$: $$f(x) + g(x) = 6 + 11 = 17$$.

Thus $$x$$ could be 32, 33, 34, 35 or 36: $$31 \lt x \lt 37$$.

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Hi bunuel I cannot understand why f(31) = 5 ?????? Can you explain it ! tks !
Math Expert V
Joined: 02 Sep 2009
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langtuprovn2007 wrote:
Hi bunuel I cannot understand why f(31) = 5 ?????? Can you explain it ! tks !

$$f(n)$$ is the number of positive perfect squares less than $$n$$, so f(31) is the number of positive perfect squares less than 31. There are 5 positive positive perfect squares less than 31: 1, 4, 9, 16, and 25. Therefore, f(31) = 5.

Hope it's clear.
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A really good-conceptual and simple yet elegant question. And that makes it the perfect GMAT Material.. _________________
The Mind is everything . What you think you become. - Lord Buddha

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Bunuel wrote:
Official Solution:

The functions $$f$$ and $$g$$ are defined for all the positive integers $$n$$ by the following rule: $$f(n)$$ is the number of positive perfect squares less than $$n$$ and $$g(n)$$ is the number of primes numbers less than $$n$$. If $$f(x) + g(x) = 16$$, then $$x$$ is in the range:

A. $$30 \lt x \lt 36$$
B. $$30 \lt x \lt 37$$
C. $$31 \lt x \lt 37$$
D. $$31 \lt x \lt 38$$
E. $$32 \lt x \lt 38$$

Positive perfect squares: 1, 4, 9, 16, 25, 36, ..,

Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ...

If $$x = 31$$, then $$f(31) = 5$$ and $$g(31) = 10$$: $$f(x) + g(x) = 5 + 10 = 15$$.

If $$x = 32$$, then $$f(32) = 5$$ and $$g(32) = 11$$: $$f(x) + g(x) = 5 + 11 = 16$$.

...

If $$x = 36$$, then $$f(36) = 5$$ and $$g(36) = 11$$: $$f(x) + g(x) = 5 + 11 = 16$$.

If $$x = 37$$, then $$f(37) = 6$$ and $$g(37) = 11$$: $$f(x) + g(x) = 6 + 11 = 17$$.

Thus $$x$$ could be 32, 33, 34, 35 or 36: $$31 \lt x \lt 37$$.

Wouldn't D also be correct, since 31<x<37 would also be included in 31<x<38, and therefore "$$x$$ is in the range" 31<x<38?

I think the question should instead be something like "If $$f(x) + g(x) = 16$$, then the exact range of possible values of x is:" or "If $$f(x) + g(x) = 16$$, then which of the following represents all possible values of x?"
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mmagyar wrote:
Bunuel wrote:
Official Solution:

The functions $$f$$ and $$g$$ are defined for all the positive integers $$n$$ by the following rule: $$f(n)$$ is the number of positive perfect squares less than $$n$$ and $$g(n)$$ is the number of primes numbers less than $$n$$. If $$f(x) + g(x) = 16$$, then $$x$$ is in the range:

A. $$30 \lt x \lt 36$$
B. $$30 \lt x \lt 37$$
C. $$31 \lt x \lt 37$$
D. $$31 \lt x \lt 38$$
E. $$32 \lt x \lt 38$$

Positive perfect squares: 1, 4, 9, 16, 25, 36, ..,

Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ...

If $$x = 31$$, then $$f(31) = 5$$ and $$g(31) = 10$$: $$f(x) + g(x) = 5 + 10 = 15$$.

If $$x = 32$$, then $$f(32) = 5$$ and $$g(32) = 11$$: $$f(x) + g(x) = 5 + 11 = 16$$.

...

If $$x = 36$$, then $$f(36) = 5$$ and $$g(36) = 11$$: $$f(x) + g(x) = 5 + 11 = 16$$.

If $$x = 37$$, then $$f(37) = 6$$ and $$g(37) = 11$$: $$f(x) + g(x) = 6 + 11 = 17$$.

Thus $$x$$ could be 32, 33, 34, 35 or 36: $$31 \lt x \lt 37$$.

Wouldn't D also be correct, since 31<x<37 would also be included in 31<x<38, and therefore "$$x$$ is in the range" 31<x<38?

I think the question should instead be something like "If $$f(x) + g(x) = 16$$, then the exact range of possible values of x is:" or "If $$f(x) + g(x) = 16$$, then which of the following represents all possible values of x?"

Hello. D can't be the answer because
(1) there are 6 perfect squares (1, 4, 9, 16, 25, 36) < 38
(2) there are 11 prime numbers < 38 (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37)
==> f(x) + g(x) could be 6 + 11 = 17, NOT 16.

You have to narrow the range in order to satisfy the equation f(x) + g(x) = 16.

Thus, D is wrong.
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pqhai wrote:
mmagyar wrote:
Bunuel wrote:
Official Solution:

The functions $$f$$ and $$g$$ are defined for all the positive integers $$n$$ by the following rule: $$f(n)$$ is the number of positive perfect squares less than $$n$$ and $$g(n)$$ is the number of primes numbers less than $$n$$. If $$f(x) + g(x) = 16$$, then $$x$$ is in the range:

A. $$30 \lt x \lt 36$$
B. $$30 \lt x \lt 37$$
C. $$31 \lt x \lt 37$$
D. $$31 \lt x \lt 38$$
E. $$32 \lt x \lt 38$$

Positive perfect squares: 1, 4, 9, 16, 25, 36, ..,

Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ...

If $$x = 31$$, then $$f(31) = 5$$ and $$g(31) = 10$$: $$f(x) + g(x) = 5 + 10 = 15$$.

If $$x = 32$$, then $$f(32) = 5$$ and $$g(32) = 11$$: $$f(x) + g(x) = 5 + 11 = 16$$.

...

If $$x = 36$$, then $$f(36) = 5$$ and $$g(36) = 11$$: $$f(x) + g(x) = 5 + 11 = 16$$.

If $$x = 37$$, then $$f(37) = 6$$ and $$g(37) = 11$$: $$f(x) + g(x) = 6 + 11 = 17$$.

Thus $$x$$ could be 32, 33, 34, 35 or 36: $$31 \lt x \lt 37$$.

Wouldn't D also be correct, since 31<x<37 would also be included in 31<x<38, and therefore "$$x$$ is in the range" 31<x<38?

I think the question should instead be something like "If $$f(x) + g(x) = 16$$, then the exact range of possible values of x is:" or "If $$f(x) + g(x) = 16$$, then which of the following represents all possible values of x?"

Hello. D can't be the answer because
(1) there are 6 perfect squares (1, 4, 9, 16, 25, 36) < 38
(2) there are 11 prime numbers < 38 (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37)
==> f(x) + g(x) could be 6 + 11 = 17, NOT 16.

You have to narrow the range in order to satisfy the equation f(x) + g(x) = 16.

Thus, D is wrong.

I think you misunderstand the unintended meaning of the question and my point. The question tells us that "$$f(x) + g(x) = 16$$" and then asks "$$x$$ is in the range:"

That last part is very important. Because it asks "x IS in the range" (emphasis added), we are basically being asked a must be true question. We don't know exactly what x is, but we need to find the answer with a range of values that definitely includes whatever x actually is.

Since we know that the range of possible x values is 32-36, x must be one of those numbers and the correct answer must include that entire range to be correct.

D says $$31 \lt x \lt 38$$. While this includes more than all of the possible values of x, it still includes all of the possible values of x. Thus, x is definitely in the range given in D, as directly asked by the question.

I completely understand the math. I completely understand the intent of the question. I got the question correct when I did it, but I noticed that other answers are also technically correct. The way the question is worded does not match the intent of the question and should be fixed.

As a general piece of advice, one should be very careful about what the GMAT asks. I often see test-takers answer a different question than what is asked or not understand how to analyze "must be true" questions.
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I think this is a high-quality question and the explanation isn't clear enough, please elaborate. What exactly are we doing here ? Are we plugging values into the function by picking one from the range in the answer choices ?
Intern  Joined: 06 Dec 2015
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I think this is a poor-quality question and the explanation isn't clear enough, please elaborate. choice 3,4,5 are all correct
Math Expert V
Joined: 02 Sep 2009
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MissMavis wrote:
I think this is a poor-quality question and the explanation isn't clear enough, please elaborate. choice 3,4,5 are all correct

The question is fine. If you don't understand the solution please be a little bit more specific when asking a question. Thank you.
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Although it took me some time to understand this question, I liked it a lot! very conceptual & realistic to GMAT! Kudos to GMAT club!
Current Student B
Joined: 08 Jan 2015
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Indeed, mmagyar is right - the appropriate question for this task is: "In what range all x satisfy f(x)+g(x)=16". Otherwise, the question itself is incorrect, since if I pick one x from a range, and it fits, then I answer the question.
Current Student B
Joined: 28 Aug 2016
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Just gotta be careful when listing all the prime numbers... I missed 19 and couldn't figure out what was wrong for the life of me. Intern  Joined: 29 Sep 2016
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mmagyar

Well, I understand your point of the need for preciseness of wording.

However, choice D and E are incorrect because now E excludes x=32 and D includes x=37.

Even though choice D overlaps with C, like you said, it introduces a new problem. This is crucial point to remember when solving not just in quant but in critical reasoning questions as well.

You must choose an answer that corrects the problem but does not introduces a new problem that compromises the solution.

The wording of the question has absolutely no problem "x is in the range..."

Thanks!
Intern  B
Joined: 03 Feb 2016
Posts: 8
GMAT 1: 690 Q49 V34 Show Tags

For that matter, B, C, and D are all correct.

Since the possible values of x are 32, 33, 34, 35, 36 and since we are concerned about the range in which these 5 values can lie, therefore

x lies in the range 30<x<37,

x lies in the range 31<x<37, and

x lies in the range 31<x<38

x even lies in 0<x<100

but not all numbers in the range 0<x<100 doesn't satisfy f(x) + g(x)= 16.

Its like the card trick- you ask a spectator to select a card and put it back in the deck. Then you pick 3 cards and ask "Is you card among these 3?"
Manager  B
Joined: 16 Jan 2017
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GMAT 1: 620 Q46 V29 Show Tags

great tricky question. Almost got it wrong as I almost forgot about 1 being a perfect square.
Intern  B
Joined: 21 May 2015
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I think this is a high-quality question and I agree with explanation.
Manager  B
Joined: 19 Jun 2017
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Got this one wrong but It is a Great question! I have so much fun doing Bunuel's questions!
Thank You
Intern  B
Joined: 13 Jun 2015
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Quite methodical, as opposed to my scrappy, mental-math oriented approach! Re M25-28   [#permalink] 04 Apr 2018, 02:37

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