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Bunuel
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If m and n are positive integers, and x = 2^m3^m, is m < n ? 12 Sep 2022, 07:33
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

75% (hard)

Question Stats:

57% (02:21) correct 43% (02:06) wrong based on 137 sessions

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If m and n are positive integers, and \(x = 2^m3^n\), is m < n ?

(1) x is divisible by 144
(2) x is not divisible by 648


 


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C
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Archit3110
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If m and n are positive integers, and x = 2^m3^m, is m < n ? 12 Sep 2022, 07:55
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given
m & n are +ve integers
\(x = 2^m3^m\),
target is m < n

#1
x is divisible by 144
144 ; 2^4 * 3^2
x can be 2^5*3^5 or 2^4*3^2
insufficient
#2
x is not divisible by 648

648 ; 2^3 * 3^4
x can be 2^2*3^4 or 2^4*3^3
insufficient
from 1&2
x has to be 2^4 * 3^2 or 2^4 * 3^3
all cases m>n
sufficient option C

Bunuel
If m and n are positive integers, and \(x = 2^m3^m\), is m < n ?

(1) x is divisible by 144
(2) x is not divisible by 648


 


Enjoy this brand new question we just created for the GMAT Club Tests.

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If m and n are positive integers, and x = 2^m3^m, is m < n ? 16 Oct 2022, 06:45
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Correct me if I am wrong. I do not see any info regarding n, would we still be able to answer the question?
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If m and n are positive integers, and x = 2^m3^m, is m < n ? 16 Oct 2022, 10:38
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Rainman91
Correct me if I am wrong. I do not see any info regarding n, would we still be able to answer the question?
Show more

Yep. Typo. Edited.


This is DEFINITELY, a hard, 700+, MAYBE EVEN 750+, question. Below is a complete, step-by-step solution.

Official Solution:


If \(m\) and \(n\) are positive integers, and \(x = 2^m3^n\), is \(m < n\) ?

This is certainly a challenging question, likely in the range of 700 or even 750 difficulty level. Find a detailed, step-by-step solution below.

(1) \(x\) is divisible by 144.

Factor 144:

\(144 = 2^4*3^2\).
The above means that \(m \geq 4\) and \(n \geq 2\). This is not enough to conclude whether m < n. For example:

If \(x = 2^4*3^4=1296\), then the answer is NO: \((m=4) = (n=4)\) (notice that 144 IS a factor of 1296).

If \(x = 2^4*3^5=3888\), then the answer is YES: \((m=4) < (n=5)\) (notice that 144 IS a factor of 3888).
Not sufficient.

(2) \(x\) is not divisible by 648

Factor 648:

\(648 = 2^3*3^4\).
The above means that either \(m < 3\) OR \(n < 4\) (to put it simply, for x not to be divisible by \(648 = 2^3*3^4\), EITHER the power of 2, which is \(m\), must be less than 3 OR the factor of 3, which is \(n\), must be less than 4 (or both)). This is not enough to conclude whether \(m < n\). For example:

If \(x = 2*3=6\), then the answer is NO: \((m=1) = (n=1)\) (notice that 6 is NOT divisible by 648).

If \(x = 2*3^2=18\), then the answer is YES: \((m=1) < (n=2)\) (notice that 18 is NOT divisible by 648).
Not sufficient.

(1)+(2) When combining, we get that \(m\) cannot be less than 3 (from (2)), because from (1) we know that \(m \geq 4\), thus \(m \geq 4\) must be true (from (1)). Now, since \(m \geq 4\), then from (2) \(n < 4\) must be true. Therefore, \(m > n\) and we get a definite NO answer to the question.

Sufficient.


Answer: C­
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If m and n are positive integers, and x = 2^m3^m, is m < n ? 29 May 2024, 20:55
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Bunuel
Rainman91
Correct me if I am wrong. I do not see any info regarding n, would we still be able to answer the question?

Yep. Typo. Edited.


This is DEFINITELY, a hard, 700+, MAYBE EVEN 750+, question. Below is a complete, step-by-step solution.

GMAT CLUB OFFICIAL EXPLANATION

If m and n are positive integers, and \(x = 2^m3^n\), is m < n ?

(1) x is divisible by 144

Factor 144:

    \(144 = 2^4*3^2\).

The above means that \(m \geq 4\) and \(n \geq 3\). This is not enough to conclude whether m < n. For example:

    If \(x = 2^4*3^4=1296\), then the answer is NO: \((m=4) = (n=4)\) (notice that 144 IS a factor of 1296).
    If \(x = 2^4*3^5=3888\), then the answer is YES: \((m=4) < (n=5)\) (notice that 144 IS a factor of 3888).

Not sufficient.

(2) x is not divisible by 648

Factor 648:

    \(648 = 2^3*3^4\).

The above means that either \(m < 3\) OR \(n < 4\) (to put it simply, for x not to be divisible by 648 = 2^3*3^4, EITHER the power of 2, which is m, must be less than 3 OR the factor of 3, which is n, must be less than 4 (or both)). This is not enough to conclude whether m < n. For example:

    If \(x = 2*3=6\), then the answer is NO: \((m=1) = (n=1)\) (notice that 6 is NOT divisible by 648).
    If \(x = 2*3^2=18\), then the answer is YES: \((m=1) < (n=2)\) (notice that 18 is NOT divisible by 648).

Not sufficient.

(1)+(2) When combining, we get that m cannot be less than 3 (from (2)), because from (1) we know that \(m \geq 4\), thus \(m \geq 4\) must be true (from (1)). Now, since \(m \geq 4\), then from (2) \(n < 4\) must be true (from (2)).

\(n < 4\) (from (2)) and \(n \geq 3\) (from (1)), gives \(n =3\).

So, we get that \(m \geq 4\) (m can be 4, 5, 7, ...) and \(n =3\). Therefore, \(m > n\) and we get a definite NO answer to the question.

Sufficient.

Answer: C.
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­
For (1) x is divisible by 144

Factor 144:

    \(144 = 2^4*3^2\).

I think \(n \geq 2\) and n does not need to be greater than 3­
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Bunuel
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If m and n are positive integers, and x = 2^m3^m, is m < n ? 30 May 2024, 00:24
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hughng92
Bunuel
Rainman91
Correct me if I am wrong. I do not see any info regarding n, would we still be able to answer the question?

Yep. Typo. Edited.


This is DEFINITELY, a hard, 700+, MAYBE EVEN 750+, question. Below is a complete, step-by-step solution.

GMAT CLUB OFFICIAL EXPLANATION

If m and n are positive integers, and \(x = 2^m3^n\), is m < n ?

(1) x is divisible by 144

Factor 144:

    \(144 = 2^4*3^2\).

The above means that \(m \geq 4\) and \(n \geq 3\). This is not enough to conclude whether m < n. For example:

    If \(x = 2^4*3^4=1296\), then the answer is NO: \((m=4) = (n=4)\) (notice that 144 IS a factor of 1296).
    If \(x = 2^4*3^5=3888\), then the answer is YES: \((m=4) < (n=5)\) (notice that 144 IS a factor of 3888).

Not sufficient.

(2) x is not divisible by 648

Factor 648:

    \(648 = 2^3*3^4\).

The above means that either \(m < 3\) OR \(n < 4\) (to put it simply, for x not to be divisible by 648 = 2^3*3^4, EITHER the power of 2, which is m, must be less than 3 OR the factor of 3, which is n, must be less than 4 (or both)). This is not enough to conclude whether m < n. For example:

    If \(x = 2*3=6\), then the answer is NO: \((m=1) = (n=1)\) (notice that 6 is NOT divisible by 648).
    If \(x = 2*3^2=18\), then the answer is YES: \((m=1) < (n=2)\) (notice that 18 is NOT divisible by 648).

Not sufficient.

(1)+(2) When combining, we get that m cannot be less than 3 (from (2)), because from (1) we know that \(m \geq 4\), thus \(m \geq 4\) must be true (from (1)). Now, since \(m \geq 4\), then from (2) \(n < 4\) must be true (from (2)).

\(n < 4\) (from (2)) and \(n \geq 3\) (from (1)), gives \(n =3\).

So, we get that \(m \geq 4\) (m can be 4, 5, 7, ...) and \(n =3\). Therefore, \(m > n\) and we get a definite NO answer to the question.

Sufficient.

Answer: C.
­
For (1) x is divisible by 144

Factor 144:

    \(144 = 2^4*3^2\).

I think \(n \geq 2\) and n does not need to be greater than 3­
Show more
­
Edited the typo. Thank you!

P.S. Worth noting though that such type of pure algebraic questions are no longer a part of the DS syllabus of the GMAT.

DS questions in GMAT Focus encompass various types of word problems, such as:

Word Problems
Work Problems
Distance Problems
Mixture Problems
Percent and Interest Problems
Overlapping Sets Problems
Statistics Problems
Combination and Probability Problems


While these questions may involve or necessitate knowledge of algebra, arithmetic, inequalities, etc., they will always be presented in the form of word problems. You won’t encounter pure "algebra" questions like, "Is x > y?" or "A positive integer n has two prime factors..."

Check GMAT Syllabus for Focus Edition

Hope it helps.
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