GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 12 Dec 2019, 20:37 ### 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

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.  # If 144/x is an integer and 108/x is an integer, which of the following

Author Message
TAGS:

### Hide Tags

Intern  Joined: 03 Jul 2009
Posts: 8
If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

4
20 00:00

Difficulty:   75% (hard)

Question Stats: 52% (01:48) correct 48% (01:52) wrong based on 404 sessions

### HideShow timer Statistics

If 144/x is an integer and 108/x is an integer, which of the following must be true?

I. 9/x is an integer.
II. 12/x is an integer.
III. 36/x is an integer.

A. I only
B. III only
C. I and II only
D. II and III only
E. I, II, and III

Originally posted by Navigator on 03 Nov 2009, 23:08.
Last edited by Bunuel on 01 Jul 2019, 00:14, edited 2 times in total.
Edited the question and added the OA
Senior Manager  B
Joined: 31 Aug 2009
Posts: 338
Location: Sydney, Australia
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

3
This is a number properties question.

There are two approaches you can take.

Proof approach:
For 144/x to be an integer X can be any multiple of the following 2,2,2,2,3,3
For 108/x to be an integer x can be any multiple of the following 2,2,3,3,3
The largest possible value for X will be the multiple of the common elements 2x2x3x3 = 36
If you know your number props well this is quick and easy

Back solve approach:
Statement I) 9/x is an integer – this is true if x<=9 and a factor of 9. So this is true when x is 3 or 9. But X could be 12 and still meet conditions. Not True.
Statement II) 12/x is an integer – this is true if x<=12 and a factor of 12. So this is true when x is 2,3,4,12. But X could be 18 and still meet conditions. Not True.
We could go and solve this (and if you do you’ll see that only 36 meets all criteria) but from the answer choices we can see that the only answer choice that does not include 1 and 2 is choice B.

I prefer the proof approach, its neater and quick if you know your number props.
Manager  Joined: 04 Nov 2009
Posts: 63
WE 1: Research
WE 2: Corporate Strat
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

1
1
from the proof approach, the smallest possible value for x is 2 and the largest is 36.

So basically we just need to test these 2 extremes. Which shows 36 only will work.
Director  Status: Can't wait for August!
Joined: 13 Sep 2011
Posts: 985
Location: United States (MA)
Concentration: Marketing, Strategy
GMAT 1: 660 Q44 V37 GMAT 2: 680 Q45 V38 GMAT 3: 710 Q45 V42 GPA: 3.32
WE: Information Technology (Retail)
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

2
36 is the greatest common factor for 144 and 108, so the greatest possible value of X = 36

so, only III is true for all values X

Intern  Joined: 03 Dec 2010
Posts: 21
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

Hi,

I didnt get this one. The q askes for which one of the following is an integer. So i plugged in nos. to see, and found that each option is has divisors. On what basis is the answer 36 as d qustn askes for "is an integer" and not greatest factor?

Pls explain,
Thnx
Math Expert V
Joined: 02 Sep 2009
Posts: 59712
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

1
1
priyalr wrote:
Hi,

I didnt get this one. The q askes for which one of the following is an integer. So i plugged in nos. to see, and found that each option is has divisors. On what basis is the answer 36 as d qustn askes for "is an integer" and not greatest factor?

Pls explain,
Thnx

If 144/x is an integer and 108/x is an integer, which of the following must be true?

I. 9/x is an integer
II. 12/x is an integer
III. 36/x is an integer

A. I only
B. III only
C. I and II only
D. II and III only
E. I, II and III

The question asks which of the following MUST be true, not COULD be true. The largest possible value of x is 36, GCD of 144 and 108, and if x=36 then ONLY III is true.

Check more Must or Could be True Questions to practice: search.php?search_id=tag&tag_id=193

Hope it helps.
_________________
Manager  Status: And the Prep starts again...
Joined: 03 Aug 2010
Posts: 96
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

Why is statement-2 to not true?

Statement II) 12/x is an integer –
this is true if x<=12 and a factor of 12. So this is true when x is 2,3,4,12. Is this not sufficient as both 144 and 108 are divisible by any of the numbers 2,3,4,12 ?
Math Expert V
Joined: 02 Sep 2009
Posts: 59712
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

3
1
ENAFEX wrote:
Why is statement-2 to not true?

Statement II) 12/x is an integer –
this is true if x<=12 and a factor of 12. So this is true when x is 2,3,4,12. Is this not sufficient as both 144 and 108 are divisible by any of the numbers 2,3,4,12 ?

If 144/x is an integer and 108/x is an integer, which of the following must be true?
I. 9/x is an integer
II. 12/x is an integer
III. 36/x is an integer

A. I only
B. III only
C. I and II only
D. II and III only
E. I, II and III

The question asks which of the following MUST be true, not COULD be true. The largest possible value of x is 36 (the greatest common factor of 144 and 108), and if x=36 then ONLY III is true.

_________________
Manager  Joined: 28 May 2009
Posts: 143
Location: United States
Concentration: Strategy, General Management
GMAT Date: 03-22-2013
GPA: 3.57
WE: Information Technology (Consulting)
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

If $$\frac{144}{x}$$ is an integer, and $$\frac{108}{x}$$ is an integer, which of the following must be true?

I. $$\frac{9}{x}$$ is an integer
II. $$\frac{12}{x}$$ is an integer
III. $$\frac{36}{x}$$ is an integer

(A) I only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II, and III

Source: Gmat Hacks 1800 set.

This is a repost, but the previous postings are still confusing. So the way I read this is "If $$\frac{144}{2}$$ is an integer and $$\frac{108}{2}$$ is an integer .." but that reasoning seems to be wrong, can someone explain why?

Edit - I get it now. It's the "MUST BE TRUE" part that I forgot to factor into. Anyway, if you get that part, this is pretty easy question.
_________________
Intern  B
Joined: 11 Jul 2013
Posts: 31
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

the way i see is x can be 1 or 2 or 3

how do we decide whether 9/x , 12/x or 36/x is an integer if i chose x as 1 still {144}/{x} is an integer, and {108}/{x} is also an integer...same happens when i chose 2 but x as 1 and x as 2 given me differnt answer choices
Math Expert V
Joined: 02 Sep 2009
Posts: 59712
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

tyagigar wrote:
the way i see is x can be 1 or 2 or 3

how do we decide whether 9/x , 12/x or 36/x is an integer if i chose x as 1 still {144}/{x} is an integer, and {108}/{x} is also an integer...same happens when i chose 2 but x as 1 and x as 2 given me differnt answer choices

x could be 1, 2, 3, 4, 6, 9, 12, 18 or 36 (these are common factors of 144 and 108). The question asks which of the options MUST be an integer. Now, only 36/x is an integer for all possible values of x.

Does this make sense?

Similar questions to practice:
if-n-is-a-positive-integer-and-n-2-is-divisible-by-96-then-127364.html
if-n-is-a-positive-integer-and-n-2-is-divisible-by-72-then-90523.html
a-certain-clock-marks-every-hour-by-striking-a-number-of-tim-91750.html
if-m-and-n-are-positive-integer-and-1800m-n3-what-is-108985.html
if-x-and-y-are-positive-integers-and-180x-y-100413.html
n-is-a-positive-integer-and-k-is-the-product-of-all-integer-104272.html
if-x-is-a-positive-integer-and-x-2-is-divisible-by-32-then-88388.html
if-n-and-y-are-positive-integers-and-450y-n-92562.html
if-5400mn-k-4-where-m-n-and-k-are-positive-integers-109284.html

Hope this helps.
_________________
Intern  Joined: 21 Apr 2014
Posts: 39
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

1
So the best way to start is to break each one into their prime factors:

144=2*2*2*2*3*3

108=2*2*3*3*3

We know that x has to be a combination of the primes such that it is a prime or combination of the primes that it both of those numbers can be evenly divided by it.

Now let's take a look at the options. These are things that MUST be true, so if we can find a scenario where they are not then we know that we can eliminate it.

I. 9/x doesn't have to be an integer because x could be 2 (This eliminates A, C and E)
II. 12/x doesn't have to be an integer because x could be 9 (This eliminates D)

Now we know that B is the only option left we can double check it

III. 36 (2*2*3*3) does have to be an integer because there is no singular or combination of primes that divides evenly into 144 and 108 and not 36.

_________________
Eliza
GMAT Tutor
bestgmatprepcourse.com
Manager  Joined: 23 Oct 2014
Posts: 84
Concentration: Marketing
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

Navigator wrote:
If 144/x is an integer and 108/x is an integer, which of the following must be true?

I. 9/x is an integer.
II. 12/x is an integer.
III. 36/x is an integer.

A. I only
B. III only
C. I and II only
D. II and III only
E. I, II, and III

So we need to find the largest number to X that is divisible by both 144 and 108.

144=3^2 x 4^2
108=3^3 x 4

So looking at this the only numbers that can be divisible is 3^2 x 4 = 36

Looking at the options.

1. 9/36 is not divisible
2. 12/36 is not divisible
3. 36/36 is divisible

Thus the only option that works is 3.
Board of Directors P
Joined: 17 Jul 2014
Posts: 2491
Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30 GPA: 3.92
WE: General Management (Transportation)
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

I attacked this question in a different way:
prime factorization of 144 = 2*2*2*2*3*3 = so we have four 4's and two of 3's
prime factorization of 108 = 2*2*3*3*3 = we have two of 2's and three of 3's.

I 9/x is an integer. well, if x is 3*3 = then yes, 9/x is an integer.
but if x is 2*2*3*3 = then 9/x is not divisible. since our question asks for must be true -> we know for sure that I is not true.
Eliminate (A) I only, (C) I and II only, and (E) I, II, and III

II 12/x is an integer
well, if x is 2*2*3 = or 2*2 or 2*3 = then yes, 12/x is an integer, but x can be 3*3*2.
since it is a must be true, we can eliminate E, and thus B is the answer.
Retired Moderator P
Joined: 17 Jun 2016
Posts: 499
Location: India
GMAT 1: 720 Q49 V39 GMAT 2: 710 Q50 V37 GPA: 3.65
WE: Engineering (Energy and Utilities)
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

1
B works fine ...

X can be 3, 4, 9, 12 or 36...
hence for any of the above values of x, only 36/x will be an integer ...
_________________
Target Test Prep Representative V
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 8692
Location: United States (CA)
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

Navigator wrote:
If 144/x is an integer and 108/x is an integer, which of the following must be true?

I. 9/x is an integer.
II. 12/x is an integer.
III. 36/x is an integer.

A. I only
B. III only
C. I and II only
D. II and III only
E. I, II, and III

We are given that 144/x is an integer and 108/x is an integer. Let’s prime factorize 144 and 108.

144 = 12 x 12 = 2^2 x 3^1 x 2^2 x 3^1 = 2^4 x 3^2

108 = 4 x 27 = 2^2 x 3^3

Thus, the largest possible value x could be is 2^2 x 3^2 = 4 x 9 = 36, which is the GCF of 144 and 108. Furthermore, x could be any of the factors of 36. Thus, of the Roman numerals, only 36/x must be an integer.

_________________

# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Senior Manager  G
Joined: 12 Sep 2017
Posts: 308
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

ScottTargetTestPrep wrote:
Navigator wrote:
If 144/x is an integer and 108/x is an integer, which of the following must be true?

I. 9/x is an integer.
II. 12/x is an integer.
III. 36/x is an integer.

A. I only
B. III only
C. I and II only
D. II and III only
E. I, II, and III

We are given that 144/x is an integer and 108/x is an integer. Let’s prime factorize 144 and 108.

144 = 12 x 12 = 2^2 x 3^1 x 2^2 x 3^1 = 2^4 x 3^2

108 = 4 x 27 = 2^2 x 3^3

Thus, the largest possible value x could be is 2^2 x 3^2 = 4 x 9 = 36, which is the GCF of 144 and 108. Furthermore, x could be any of the factors of 36. Thus, of the Roman numerals, only 36/x must be an integer.

ScottTargetTestPrep

Could,´t be x=1?

I just have that question-based in your above explanation.

Kind regards!
Target Test Prep Representative V
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 8692
Location: United States (CA)
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

1
jfranciscocuencag wrote:
ScottTargetTestPrep wrote:
Navigator wrote:
If 144/x is an integer and 108/x is an integer, which of the following must be true?

I. 9/x is an integer.
II. 12/x is an integer.
III. 36/x is an integer.

A. I only
B. III only
C. I and II only
D. II and III only
E. I, II, and III

We are given that 144/x is an integer and 108/x is an integer. Let’s prime factorize 144 and 108.

144 = 12 x 12 = 2^2 x 3^1 x 2^2 x 3^1 = 2^4 x 3^2

108 = 4 x 27 = 2^2 x 3^3

Thus, the largest possible value x could be is 2^2 x 3^2 = 4 x 9 = 36, which is the GCF of 144 and 108. Furthermore, x could be any of the factors of 36. Thus, of the Roman numerals, only 36/x must be an integer.

ScottTargetTestPrep

Could,´t be x=1?

I just have that question-based in your above explanation.

Kind regards!

Yes, x could be 1; however, we care about what MUST be true. That is why we immediately started with the largest possible value of x, which is 36. By doing so, we immediately see that 12/x does not have to be an integer, nor does 9/x, right?
_________________

# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Non-Human User Joined: 09 Sep 2013
Posts: 13740
Re: If 144/x is an integer and 108/x is an integer, which of the following  [#permalink]

### Show Tags

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________ Re: If 144/x is an integer and 108/x is an integer, which of the following   [#permalink] 11 Feb 2019, 16:09
Display posts from previous: Sort by

# If 144/x is an integer and 108/x is an integer, which of the following  