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Math Expert V
Joined: 02 Sep 2009
Posts: 56275

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15 00:00

Difficulty:   75% (hard)

Question Stats: 49% (01:49) correct 51% (02:17) wrong based on 135 sessions

### HideShow timer Statistics If the least common multiple of a positive integer $$x$$, $$4^3$$ and $$6^5$$ is $$6^6$$. Then $$x$$ can take how many values?

A. $$1$$
B. $$6$$
C. $$7$$
D. $$30$$
E. $$36$$

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Math Expert V
Joined: 02 Sep 2009
Posts: 56275

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3
3
Official Solution:

If the least common multiple of a positive integer $$x$$, $$4^3$$ and $$6^5$$ is $$6^6$$. Then $$x$$ can take how many values?

A. $$1$$
B. $$6$$
C. $$7$$
D. $$30$$
E. $$36$$

We are given that $$6^6=2^{6}*3^{6}$$ is the least common multiple of the following three numbers:

$$x$$;

$$4^3=2^6$$;

$$6^5 = 2^{5}*3^5$$;

First notice that $$x$$ cannot have any other primes other than 2 or/and 3, because LCM contains only these primes.

Now, since the power of 3 in LCM is higher than the powers of 3 in either the second number or in the third, than $$x$$ must have $$3^{6}$$ as its multiple (else how $$3^{6}$$ would appear in LCM?).

Next, $$x$$ can have 2 as its prime in ANY power ranging from 0 to 6, inclusive (it cannot have higher power of 2 since LCM limits the power of 2 to 6).

Thus, $$x$$ could take total of 7 values.

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Intern  Joined: 17 Jun 2014
Posts: 18

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Hi Bunuel I do not understand why 7 but not 8 ?
according to my count, it is supposed to be 3^6, 2^0, 2^1, 2^2, 2^3, 2^4, 2^5, 2^6 ????? I do not get it ! Can you explain it for me ! many thanks !
Math Expert V
Joined: 02 Sep 2009
Posts: 56275

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langtuprovn2007 wrote:
Hi Bunuel I do not understand why 7 but not 8 ?
according to my count, it is supposed to be 3^6, 2^0, 2^1, 2^2, 2^3, 2^4, 2^5, 2^6 ????? I do not get it ! Can you explain it for me ! many thanks !

x could be:

$$3^6$$;
$$2*3^6$$;
$$2^2*3^6$$;
$$2^3*3^6$$;
$$2^4*3^6$$;
$$2^5*3^6$$;
$$2^6*3^6$$.

7 values!
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Intern  Joined: 17 Jun 2014
Posts: 18

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Woa !, Why did not I think about that ! Tks Bunuel my head was just mixed up when solving this problem !!!
Senior Manager  Joined: 31 Mar 2016
Posts: 376
Location: India
Concentration: Operations, Finance
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I think this is a high-quality question and I agree with explanation.
Intern  B
Joined: 22 May 2015
Posts: 2
Location: Australia
Schools: IMD Jan'18
GMAT 1: 620 Q44 V22 GPA: 3.11

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I did this as follows

4^3 = 2^6
6^5 = 2^5 & 3^5

6^6 = 2^6 3^6

So, 2^6 is fully deductible from 6^6
3^5 is fully deductible from 6^6

So the leftover values are 2^5, 3 , x

So should the answer be 6 or 7?

I chose 6
Math Expert V
Joined: 02 Sep 2009
Posts: 56275

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bloomington wrote:
I did this as follows

4^3 = 2^6
6^5 = 2^5 & 3^5

6^6 = 2^6 3^6

So, 2^6 is fully deductible from 6^6
3^5 is fully deductible from 6^6

So the leftover values are 2^5, 3 , x

So should the answer be 6 or 7?

I chose 6

x could be:

$$3^6$$;
$$2*3^6$$;
$$2^2*3^6$$;
$$2^3*3^6$$;
$$2^4*3^6$$;
$$2^5*3^6$$;
$$2^6*3^6$$.

7 values!
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Intern  Joined: 05 Mar 2015
Posts: 1

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to be the least common multipler doesnt mean that 6^6 is the greatest number?
why can x be 2^2*6^6?

thank you
Math Expert V
Joined: 02 Sep 2009
Posts: 56275

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arthurhernandes wrote:
to be the least common multipler doesnt mean that 6^6 is the greatest number?
why can x be 2^2*6^6?

thank you

x cannot be 2^2*6^6. Where does it say that it can?

x could be:

$$3^6$$;
$$2*3^6$$;
$$2^2*3^6$$;
$$2^3*3^6$$;
$$2^4*3^6$$;
$$2^5*3^6$$;
$$2^6*3^6$$.

7 values!
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Status: GMAT - Pulling Quant and Verbal together
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Location: United States (OH)
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Hi Bunuel , I don't understand even the beginning of the solution.

We are given $$6^{6}$$ = $$4^{4}$$ and $$6^{5}$$. I understand we can change $$4^{4}$$ to $$2^{6}$$, but how did you change $$6^{5}$$ into $$3^{6}$$?
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Math Expert V
Joined: 02 Sep 2009
Posts: 56275

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msurls wrote:
Hi Bunuel , I don't understand even the beginning of the solution.

We are given $$6^{6}$$ = $$4^{4}$$ and $$6^{5}$$. I understand we can change $$4^{4}$$ to $$2^{6}$$, but how did you change $$6^{5}$$ into $$3^{6}$$?

Nothing of that kin in the question. $$4^4 = (2^2)^4=2^8$$, not 2^6, which is 4^3. Please re-read the solution carefully and read the discussion.

Hope it helps.
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Intern  B
Joined: 12 Mar 2018
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Location: Iran (Islamic Republic of)
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WE: Management Consulting (Energy and Utilities)

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I don't see any reason why this question must be marked as hard(95%)!
Maybe medium (at most 65%) would be a better indicator :/
Intern  B
Joined: 28 Apr 2018
Posts: 23

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I have doubt about the solution because the values could be :
2^0*3^6(in this case the HCF of all 3 numbers is 1)
2^1*3^6
2^2*3^6
2^3*3^6
2^4*3^6
2^5*3^6
2^6*3^6

2^0*3^5, 2^1*3^5, ...., 2^6*3^5

2^0*3^4, 2^1*3^4, ..... ,2^6*3^4

2^0*3^3, 2^1*3^3, ..... ,2^6*3^3

2^0*3^2, 2^1*3^2, ..... ,2^6*3^2

2^0*3^1, 2^1*3^1, ..... ,2^6*3^1

2^0*3^0(in this case the HCF of all 3 numbers is 1), 2^1*3^0, ..... ,2^6*3^0

Hence the values are 7*7 = 49
Math Expert V
Joined: 02 Sep 2009
Posts: 56275

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1
Godsent wrote:
I have doubt about the solution because the values could be :
2^0*3^6(in this case the HCF of all 3 numbers is 1)
2^1*3^6
2^2*3^6
2^3*3^6
2^4*3^6
2^5*3^6
2^6*3^6

2^0*3^5, 2^1*3^5, ...., 2^6*3^5

2^0*3^4, 2^1*3^4, ..... ,2^6*3^4

2^0*3^3, 2^1*3^3, ..... ,2^6*3^3

2^0*3^2, 2^1*3^2, ..... ,2^6*3^2

2^0*3^1, 2^1*3^1, ..... ,2^6*3^1

2^0*3^0(in this case the HCF of all 3 numbers is 1), 2^1*3^0, ..... ,2^6*3^0

Hence the values are 7*7 = 49

x could be:

$$3^6$$;
$$2*3^6$$;
$$2^2*3^6$$;
$$2^3*3^6$$;
$$2^4*3^6$$;
$$2^5*3^6$$;
$$2^6*3^6$$.

7 values!
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Intern  B
Joined: 04 Apr 2017
Posts: 24

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Why we couldn't take 1 as a value for X?
Target Test Prep Representative D
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 6923
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Hi HisHo,

Because if x = 1, then the lcm of 1, 4^3 and 6^5 will not be 6^6.

Posted from my mobile device
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Manager  G
Joined: 22 Jun 2017
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Location: Argentina
Schools: HBS, Stanford, Wharton
GMAT 1: 630 Q43 V34 ### Show Tags

I think this is a poor-quality question and I agree with explanation. I agree with the explanation. But I didn't understand the question in the test. And I took me several minutes to understand the question even having read the explanation before. I think that the question should be rewritten.
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Math Expert V
Joined: 02 Sep 2009
Posts: 56275

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patto wrote:
I think this is a poor-quality question and I agree with explanation. I agree with the explanation. But I didn't understand the question in the test. And I took me several minutes to understand the question even having read the explanation before. I think that the question should be rewritten.

The fact that you did not understood the question does not mean that it's a poor quality question. The question is grammatically and mathematically correct.
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GMAT 1: 560 Q41 V26 GMAT 2: 550 Q43 V23 GMAT 3: 650 Q47 V33 GPA: 2.61
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If the least common multiple of a positive integer $$x$$, $$4^3$$ and $$6^5$$ is $$6^6$$. Then $$x$$ can take how many values?

We determine the LCM by counting the highest number of powers of a given prime that is repeated across the set * multiplied by non-repeated powers

If the LCM is $$6^6$$ then only the primes 2 and 3 are present and the highest power of 3 must be contained within $$x$$ itself since:
$$4^3 = 2^6$$ does not contain $$3^6$$
and
$$6^5 = (3^5 * 2^5)$$ does not contain $$3^6$$

Thus x must contain 3^6

1 solution is $$3^6$$ (as determined already)..but what else can x contain?
X can contain contain primes found within the LCM, so it can contain multiples of 2 up to the upper limit, which is given by the LCM of $$6^6 (2^6)$$
Solution 2 = $$3^6 * 2^1$$
Solution 3 = $$3^6 * 2^2$$
Solution 4 = $$3^6 * 2^3$$
Solution 5 = $$3^6 * 2^4$$
Solution 6 = $$3^6 * 2^5$$
Solution 7 = $$3^6 * 2^6$$
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+1 Kudos if you like my post pls! M28-33   [#permalink] 28 Apr 2019, 20:40

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# M28-33

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