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

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If 144/x is an integer and 108/x is an integer, which of [#permalink] New post 03 Nov 2009, 22:08
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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
[Reveal] Spoiler: OA

Last edited by Bunuel on 16 Apr 2012, 03:59, edited 1 time in total.
Edited the question and added the OA
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Re: Number properties: Which of the following must be true? [#permalink] New post 16 Apr 2012, 04:01
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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.

Answer: B.
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Re: Number properties: Which of the following must be true? [#permalink] New post 03 Nov 2009, 22:38
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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:
Looking at the answer options
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.
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Re: Number properties: Which of the following must be true? [#permalink] New post 04 Nov 2009, 10:01
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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.
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Re: Number properties: Which of the following must be true? [#permalink] New post 04 Nov 2009, 08:45
With the proof approach you outlined, why wouldn't 9 and 12 be possible options for the answer? Aren't they both divisible by 36? Or is it that you strictly look for the multiples vs. the factors of 36?
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Re: Number properties: Which of the following must be true? [#permalink] New post 04 Nov 2009, 11:00
Great, I get it now. Thanks Yangsta, Jade3 and 4Test 1.
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Re: Number properties: Which of the following must be true? [#permalink] New post 04 Nov 2009, 12:52
I did it through backsolving approach I guess,
and it took me definitely less than 2 minutes to figure it out

But, I think better option would be to use number properties approach if u are good at that.
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Re: Number properties: Which of the following must be true? [#permalink] New post 04 Nov 2009, 13:31
mirzohidjon wrote:
I did it through backsolving approach I guess,
and it took me definitely less than 2 minutes to figure it out

But, I think better option would be to use number properties approach if u are good at that.


absolutely.... I would say that you should go with what works for you.
In this specific problem with only 3 answers its easy to test each number..... but if you had to test 5 and the numbers were larger/more difficult to deal with it can be harder.
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Re: Number properties: Which of the following must be true? [#permalink] New post 16 Apr 2012, 03:52
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 ?
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Re: If 144/x is an integer and 108/x is an integer, which of [#permalink] New post 19 Dec 2013, 23:43
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Re: If 144/x is an integer and 108/x is an integer, which of [#permalink] New post 13 Nov 2014, 14:41
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.

Thus the answer is B
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If 144/x is an integer and 108/x is an integer, which of [#permalink] New post 13 Nov 2014, 14:55
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.
If 144/x is an integer and 108/x is an integer, which of   [#permalink] 13 Nov 2014, 14:55
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