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# What is the smallest positive integer x, such that 1,152x is a perfect

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Math Expert
Joined: 02 Sep 2009
Posts: 50039
What is the smallest positive integer x, such that 1,152x is a perfect  [#permalink]

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28 Apr 2016, 04:54
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55% (hard)

Question Stats:

61% (01:45) correct 39% (01:50) wrong based on 193 sessions

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What is the smallest positive integer x, such that 1,152x is a perfect cube?

A. 4
B. 6
C. 8
D. 12
E. 18

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Joined: 06 Nov 2014
Posts: 1883
Re: What is the smallest positive integer x, such that 1,152x is a perfect  [#permalink]

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29 Apr 2016, 00:20
1
Bunuel wrote:
What is the smallest positive integer x, such that 1,152x is a perfect cube?

A. 4
B. 6
C. 8
D. 12
E. 18

We need to make 1152x a perfect cube, hence we need to have the powers a multiple of 3
1152 = 2^7*3^2

The minimum value of x for which 1152x is a perfect cube = 2^2*3 = 12

Correct Option: D
Manager
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Posts: 60
Location: India
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Re: What is the smallest positive integer x, such that 1,152x is a perfect  [#permalink]

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30 Apr 2016, 02:32
Bunuel wrote:
What is the smallest positive integer x, such that 1,152x is a perfect cube?

A. 4
B. 6
C. 8
D. 12
E. 18

Take out the factors of 1152 that will come 2*2*2*2*2*2*2*3*3. for perfect cube you need every no. raise to the power 3. for 1,152x to be a perfect cube, you need two 2 and 1 3 that means 12.
Director
Joined: 24 Nov 2015
Posts: 522
Location: United States (LA)
What is the smallest positive integer x, such that 1,152x is a perfect  [#permalink]

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25 Jun 2016, 04:00
2
prime factorization 0f 1152 = 2*2*2*2*2*2*2*3*3 = $$2^{7}$$ *$$3^2$$
for a number to be perfect cube it should have powers of its prime factors in multiple of 3 (3,6 ,9 ...etc)
we need $$2^2$$ *3 = 12 to make the given number a perfect cube so x =12
Director
Joined: 20 Feb 2015
Posts: 793
Concentration: Strategy, General Management
Re: What is the smallest positive integer x, such that 1,152x is a perfect  [#permalink]

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27 Jun 2016, 02:48
1152 =2^7*3^2
for it to be a perfect cube we have to multiply it with 2^2 *3 = 12
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Re: What is the smallest positive integer x, such that 1,152x is a perfect  [#permalink]

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20 Jul 2016, 07:58
1
1,152 is 2^7*3^2

Or 2^3 * 2^3 *2*3^2

We need 2^2 & 3, to make 1152 a perfect cube.
So 4*3=12 D.

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Manager
Joined: 20 Mar 2015
Posts: 64
Location: United States
Concentration: General Management, Strategy
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Re: What is the smallest positive integer x, such that 1,152x is a perfect  [#permalink]

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20 Jul 2016, 09:03
Bunuel wrote:
What is the smallest positive integer x, such that 1,152x is a perfect cube?

A. 4
B. 6
C. 8
D. 12
E. 18

It seems the question is asking if x is an integer placed in front of the number making it either a 5 digit perfect cube or a six digit perfect cube.
please post questions with clarity. The language here should have been : what should be the minimum value of integer x when multiplied by 1152 yields a perfect cube?
Thanks!
Math Expert
Joined: 02 Sep 2009
Posts: 50039
Re: What is the smallest positive integer x, such that 1,152x is a perfect  [#permalink]

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20 Jul 2016, 09:15
RakeshThakur wrote:
Bunuel wrote:
What is the smallest positive integer x, such that 1,152x is a perfect cube?

A. 4
B. 6
C. 8
D. 12
E. 18

It seems the question is asking if x is an integer placed in front of the number making it either a 5 digit perfect cube or a six digit perfect cube.
please post questions with clarity. The language here should have been : what should be the minimum value of integer x when multiplied by 1152 yields a perfect cube?
Thanks!

1,152x means 1,152*x.
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Re: What is the smallest positive integer x, such that 1,152x is a perfect  [#permalink]

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29 Nov 2016, 20:31
1152 = (2^7)(3^2)

To get a perfect cube, we need to get the exponents (listed above) so that they are multiples of 3 --> closest multiple for 2^7 is 2^9 --> we need 2^2 to do this (KEEP)

Let's do the same for 3^2: closest multiple is 3^3 --> we need 3 to do this (KEEP)

X should have both 2^2 & 3 in order to transform 1152 into a perfect cube.

Thus, D is correct.
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Joined: 12 Sep 2015
Posts: 3024
Re: What is the smallest positive integer x, such that 1,152x is a perfect  [#permalink]

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02 Dec 2017, 15:51
1
Top Contributor
Bunuel wrote:
What is the smallest positive integer x, such that 1,152x is a perfect cube?

A. 4
B. 6
C. 8
D. 12
E. 18

------ASIDE--------------------------------------
Let's examine a perfect CUBE.

1000 is a perfect cube since 1000 = 10^3
Now check out the prime factorization of 1000
1000 = (2)(2)(2)(5)(5)(5)
= [(2)(5)][(2)(5)][(2)(5)]
Notice that we can take the prime factorization of 1000 and divide the prime factors into THREE identical groups
-----NOW ONTO THE QUESTION------------------------

1,152 = (2)(2)(2)(2)(2)(2)(2)(3)(3)
So, 1152x = (2)(2)(2)(2)(2)(2)(2)(3)(3)(x)

A. x = 4
We get: 1152x = (2)(2)(2)(2)(2)(2)(2)(3)(3)(4)
= (2)(2)(2)(2)(2)(2)(2)(3)(3)(2)(2)
In order for the above to be a perfect CUBE, we must be able to divide the prime factors into THREE identical groups.
Since we cannot do that here, we can ELIMINATE A

B. x = 6
We get: 1152x = (2)(2)(2)(2)(2)(2)(2)(3)(3)(6)
= (2)(2)(2)(2)(2)(2)(2)(3)(3)(2)(3)
Can we divide the above prime factors into THREE identical groups?
NO! ELIMINATE B

C. x = 8
We get: 1152x = (2)(2)(2)(2)(2)(2)(2)(3)(3)(8)
= (2)(2)(2)(2)(2)(2)(2)(3)(3)(2)(2)(2)
Can we divide the above prime factors into THREE identical groups?
NO! ELIMINATE C

D. x = 12
We get: 1152x = (2)(2)(2)(2)(2)(2)(2)(3)(3)(12)
= (2)(2)(2)(2)(2)(2)(2)(3)(3)(2)(2)(3)
Can we divide the above prime factors into THREE identical groups?
YES!
1152x = [(2)(2)(2)(3)][(2)(2)(2)(3)][(2)(2)(2)(3)]
So, if x = 12, then 1152x IS a perfect cube.

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Re: What is the smallest positive integer x, such that 1,152x is a perfect &nbs [#permalink] 02 Dec 2017, 15:51
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