What is the smallest positive integer x, such that 1,152x is perfect cube?
A. 4
B. 6
C. 8
D. 12
E. 18
The OA is D.
I'm really confused with this PS question. Please, can any expert assist me with it? Thanks in advanced.
What is the smallest positive integer x, such that...
This topic has expert replies
-
- Moderator
- Posts: 2209
- Joined: Sun Oct 15, 2017 1:50 pm
- Followed by:6 members
GMAT/MBA Expert
- Brent@GMATPrepNow
- GMAT Instructor
- Posts: 16207
- Joined: Mon Dec 08, 2008 6:26 pm
- Location: Vancouver, BC
- Thanked: 5254 times
- Followed by:1268 members
- GMAT Score:770
------ASIDE--------------------------------------LUANDATO wrote:What is the smallest positive integer x, such that 1,152x is perfect cube?
A. 4
B. 6
C. 8
D. 12
E. 18
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)
Let's test the answer choices....
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.
Answer: D