It is currently 10 Dec 2017, 23:00

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

If a and b are integers, and 240a = b 3 , which of the following must

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Manager
Manager
User avatar
G
Joined: 07 Jun 2017
Posts: 176

Kudos [?]: 100 [0], given: 59

Location: India
Concentration: Technology, General Management
GMAT 1: 660 Q46 V38
GPA: 3.6
WE: Information Technology (Computer Software)
GMAT ToolKit User
If a and b are integers, and 240a = b 3 , which of the following must [#permalink]

Show Tags

New post 26 Oct 2017, 22:28
2
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

52% (01:26) correct 48% (01:43) wrong based on 75 sessions

HideShow timer Statistics

If \(a\) and \(b\) are integers, and \(240a = b^3\), which of the following must be an integer?

I. \(\frac{a}{300}\)

II. \(\frac{a}{600}\)

III.\(\frac{a}{450}\)

A. None
B. I only
C. III only
D. I and III only
E. I, II, and II
[Reveal] Spoiler: OA

_________________

Regards,
Naveen
email: nkmungila@gmail.com
Please press kudos if you like this post

Kudos [?]: 100 [0], given: 59

1 KUDOS received
Manager
Manager
User avatar
S
Joined: 17 Oct 2016
Posts: 148

Kudos [?]: 38 [1], given: 89

Location: India
Concentration: Operations, Strategy
GPA: 3.7
WE: Design (Real Estate)
GMAT ToolKit User Premium Member
Re: If a and b are integers, and 240a = b 3 , which of the following must [#permalink]

Show Tags

New post 26 Oct 2017, 22:32
1
This post received
KUDOS
Option D
a shall be 2^2*5^2*3^2

Both I and III fits this. Hence I and III are proven. Option D

Posted from my mobile device
_________________

Help with kudos if u found the post useful. Thanks

Kudos [?]: 38 [1], given: 89

1 KUDOS received
Senior Manager
Senior Manager
User avatar
G
Joined: 02 Jul 2017
Posts: 281

Kudos [?]: 106 [1], given: 66

GMAT 1: 730 Q50 V38
GMAT ToolKit User CAT Tests
Re: If a and b are integers, and 240a = b 3 , which of the following must [#permalink]

Show Tags

New post 27 Oct 2017, 14:48
1
This post received
KUDOS
Given : \(240*a = b^3\) and a and b are integers.

as \(240 = 2^4 * 3* 5\)

For b to be an integer and given \(b^3= 2^4 * 3* 5 *a\) => value of a should be at least => \(a= 2^2*3^2*5^2\) to form cube of a number.

so minimum value of a = \(a= 2^2*3^2*5^2 = 900\). And now "a" can be a multiple of this number with any other cube of an integer
=> a= 900x => where x is cube of an integer

lets check given options now

1. \(\frac{a}{300} = \frac{900x}{300} = 3x\) => integer True
2. \(\frac{a}{600} = \frac{900x}{600} = \frac{3x}{2}\) => may or may not be an integer
3. \(\frac{a}{450} = \frac{900x}{450} = 2x\) => integer True


Answer: D

Kudos [?]: 106 [1], given: 66

1 KUDOS received
Intern
Intern
avatar
B
Joined: 19 Jun 2017
Posts: 36

Kudos [?]: 7 [1], given: 27

GMAT 1: 660 Q39 V40
GMAT ToolKit User
If a and b are integers, and 240a = b 3 , which of the following must [#permalink]

Show Tags

New post 28 Oct 2017, 09:51
1
This post received
KUDOS
Hi Nikkb,

I understood that we need to break 240 down into its primes, but I'm having a hard time trying to figure out where to go from there; particularly this part in your working:



Nikkb wrote:

For b to be an integer and given \(b^3= 2^4 * 3* 5 *a\) => value of a should be at least => \(a= 2^2*3^2*5^2\) to form cube of a number.


Answer: D



Thanks in advance!

Kudos [?]: 7 [1], given: 27

2 KUDOS received
Director
Director
avatar
P
Joined: 25 Feb 2013
Posts: 613

Kudos [?]: 302 [2], given: 38

Location: India
GPA: 3.82
GMAT ToolKit User Reviews Badge
If a and b are integers, and 240a = b 3 , which of the following must [#permalink]

Show Tags

New post 28 Oct 2017, 10:08
2
This post received
KUDOS
calappa1234 wrote:
Hi Nikkb,

I understood that we need to break 240 down into its primes, but I'm having a hard time trying to figure out where to go from there; particularly this part in your working:



Nikkb wrote:

For b to be an integer and given \(b^3= 2^4 * 3* 5 *a\) => value of a should be at least => \(a= 2^2*3^2*5^2\) to form cube of a number.


Answer: D



Thanks in advance!


Hi calappa1234

it is mentioned that \(b\) is an integer so \(b^3\) must be an integer

as \(b^3=2^4*3*5*a => b=(2^4*3*5*a)^{\frac{1}{3}}\)

for \(b\) to be an integer \(a\) has to be of the form \(= 2^2*3^2*5^2*k^3\), where \(k\) is any integer

so \(a=900k^3\)

Kudos [?]: 302 [2], given: 38

2 KUDOS received
Senior Manager
Senior Manager
User avatar
G
Joined: 02 Jul 2017
Posts: 281

Kudos [?]: 106 [2], given: 66

GMAT 1: 730 Q50 V38
GMAT ToolKit User CAT Tests
If a and b are integers, and 240a = b 3 , which of the following must [#permalink]

Show Tags

New post 28 Oct 2017, 13:47
2
This post received
KUDOS
calappa1234 wrote:
Hi Nikkb,
I understood that we need to break 240 down into its primes, but I'm having a hard time trying to figure out where to go from there; particularly this part in your working:
Nikkb wrote:
For b to be an integer and given \(b^3= 2^4 * 3* 5 *a\) => value of a should be at least => \(a= 2^2*3^2*5^2\) to form cube of a number.
Answer: D

Thanks in advance!



We have \(b^3= 2^4 * 3* 5 *a\)

=>\(b = \sqrt[3] 2^4 * 3* 5 *a = 2 \sqrt[3] 2 * 3* 5 *a\)

Now as b is an integer, above cube root value need to be an integer. to make it an integer we should have cube of all the integers present inside the root.

so minimum value of "a" required to make "b" as an integer = \(2^2*3^2*5^2 = 900\)

this will be minimum value of a . other possible values of "a" => above value * cube of an integer => as b need to be an integer so we need to have cubes of integers inside the root.

so we can write \(a = 2^2*3^2*5^2* x\) , where x is cube of some integer.

=> we can write \(a = 900x\) , where x is cube of some integer.

Kudos [?]: 106 [2], given: 66

1 KUDOS received
VP
VP
avatar
P
Joined: 22 May 2016
Posts: 1104

Kudos [?]: 392 [1], given: 639

If a and b are integers, and 240a = b 3 , which of the following must [#permalink]

Show Tags

New post 28 Oct 2017, 15:27
1
This post received
KUDOS
1
This post was
BOOKMARKED
nkmungila wrote:
If \(a\) and \(b\) are integers, and \(240a = b^3\), which of the following must be an integer?

I. \(\frac{a}{300}\)

II. \(\frac{a}{600}\)

III.\(\frac{a}{450}\)

A. None
B. I only
C. III only
D. I and III only
E. I, II, and II

If \(240a = b^3\), where \(a\) and \(b\) are integers and \(b^3\) is a perfect cube, then \(240a\) is also a perfect cube.

We must find prime factors to make \(240a\) a perfect cube.

The term \(240a\) must contain "triplets" of prime factors, that is, prime factors in groups of three.

A perfect square, for example, must have prime factors that are paired evenly, that can be grouped in twos or couplets.
Thus, perfect square \(144 = (2^43^2) = 2^22^23^2\)

Similarly, a perfect cube's prime factors must be in groups of three, or in triplets. Thus \(1000 = 2^35^3\)

\(240a\) must contain enough factors to form the cube of an integer. But 240 is not a perfect cube; so \(a\) must contain the "missing" perfect cube factors.

Find the prime factors that 240 is "missing" to be a perfect cube. Multiply those missing factors. That product = least value of a. Then assess.

1) Prime factorize: \(240 = 2^43^15^1\): None of the factors is in a group of three.

2) Calculate what is needed to make each factor a triplet. That tells you what \(a\) must be at a minimum*:

\(240 = 2^43^15^1\). Each factor lacks two copies of itself to create a triplet.

--We need \(2^2\) to get a triplet \(2^6\)
--We need \(3^2\) to get a triplet \(3^3\)
--We need \(5^2\) to get a triplet \(5^3\)

So we need \(2^23^25^2 = a = 900\)

3) No need to multiply 900 by 240. The answers ask only about \(a\). Assess: Which must be integers?

I. \(\frac{a}{300}\)
YES. 900 is the minimum value of \(a\). See notes below
900 divided by ANY factor of itself will be an integer. \(\frac{900}{300} = 3\) KEEP

II. \(\frac{a}{600}\)
NO. We need only ONE "not integer" to reject "must be integer."
600 is not a factor of 900. And \(\frac{900}{600} = 1.5\) REJECT

III.\(\frac{a}{450}\)
YES. 450 is a factor of 900. KEEP.

ANSWER D, I and III only

*At a "minimum"? \(2^23^25^2\) are the minimum missing factors needed to create a perfect cube.
But we do not know how great the actual number is (240a and \(b^3\)).

a could contain another perfect cube as a factor.
As long as the factors are (240)(a)(C), where (C) is a perfect cube, the equation will work.

Example: Let (C) = (1), a perfect cube.
So (240)(900)(1) = 216,000. Perfect cube.
216,000 = 60\(^3\). Maybe \(b^3 = 60^3\)

But let (C) = (8), also a perfect cube.
(240)(900)(8) = 1,728,000 = 120\(^3\)
So maybe \(b^3 = 120^3\)

Thus \(2^23^25^2 = 900 =\) the least value of a.

Kudos [?]: 392 [1], given: 639

If a and b are integers, and 240a = b 3 , which of the following must   [#permalink] 28 Oct 2017, 15:27
Display posts from previous: Sort by

If a and b are integers, and 240a = b 3 , which of the following must

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


GMAT Club MBA Forum Home| About| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.