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Division Question Driving Me Crazy

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False proof way to solve number property questions like [#permalink] New post 09 Jan 2013, 19:42
What is the false proof way of solving questions like, 'If j is divisible by 12 and 10, is j divisible by 24?'
(a) Yes
(b) No
(c) Cannot be determined

How I happened to solve this question is:
2,2,3 are factors of 12.
2,5 are factors of 10.

Since 24 is 8*3 which can be formed from above factors of 12 and 10(as per factor foundation rule), j is divisible by 24.

But, the answer is 'Cannot be determined', since he rightly shows that if j=60 then 60 is not divisible by 24, where as it is divisible by 12, 10.
Thus, can anyone show me a false proof procedure(without falling into any traps as above), to solve the above type of questions?
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Re: False proof way to solve number property questions like [#permalink] New post 09 Jan 2013, 20:05
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As j is divisible by 12 &10, you will need to find the LCM(12,10) and check if 24 is factor of that LCM.

In this case LCM(12,10) = 60 and 24 is not factor of 60 -> Hence j is not always divisible by 24. (Hence answer -> cannot be determined)

Lets take another example, say K is divisible by 30 & 8 -> LCM(30,8)=120 -> 24 is factor of 120 -> Hence K is always divisible by 24.
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Re: False proof way to solve number property questions like [#permalink] New post 09 Jan 2013, 20:13
PraPon wrote:
As j is divisible by 12 &10, you will need to find the LCM(12,10) and check if 24 is factor of that LCM.

In this case LCM(12,10) = 60 and 24 is not factor of 60 -> Hence j is not always divisible by 24. (Hence answer -> cannot be determined)

Lets take another example, say K is divisible by 30 & 8 -> LCM(30,8)=120 -> 24 is factor of 120 -> Hence K is always divisible by 24.



Solid answer. Something I was looking for. Kudos!
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Division Question Driving Me Crazy [#permalink] New post 27 Apr 2013, 21:06
Hi guys: This question is from the Number Properties book (5th ed) from Manhattan Guide. My answer is different from that stated in the solutions, so I am wondering if the book is incorrect.

So here is the question. I greatly appreciate the help as I'm about to smash my head into the wall. Thanks!

For questions #1-6, answer each question YES, NO, or CANNOT BE DETERMINED. If your answer
is CANNOT BE DETERMINED, use two numerical examples to show how the problem could
go either way. All variables in problems #1-6 are assumed to be integers.

4. If j is divisible by 12 and 10, is j divisible by 24?
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Re: Division Question Driving Me Crazy [#permalink] New post 27 Apr 2013, 21:49
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12=2*2*3
10=2*5

J is divisible by both so \frac{j}{2*2*3*5}=integer, is j divisible by 24=2*2*2*3?

maybe
example j=2*2*3*5*7= 240 divisible by 10 and 12 but NOT by 24
example j=2*2*3*5*11=660 divisible by 10 and 12 but NOT by 24
example j=2*2*3*5*2=120 divisivle by 10 and 12 AND 24

Hope this clarifies, let me know
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Re: Division Question Driving Me Crazy [#permalink] New post 27 Apr 2013, 23:20
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tmipanthers wrote:
Hi guys: This question is from the Number Properties book (5th ed) from Manhattan Guide. My answer is different from that stated in the solutions, so I am wondering if the book is incorrect.

So here is the question. I greatly appreciate the help as I'm about to smash my head into the wall. Thanks!

For questions #1-6, answer each question YES, NO, or CANNOT BE DETERMINED. If your answer
is CANNOT BE DETERMINED, use two numerical examples to show how the problem could
go either way. All variables in problems #1-6 are assumed to be integers.

4. If j is divisible by 12 and 10, is j divisible by 24?


Keep in mind that when you say a is divisible by b, it means 'b' is a factor of 'a'.

i.e. a = n*b
If b = 12, a can be 12/24/36/48/60.... etc

If j is divisible by 12, it means j has 12 as a factor (so j has at least two 2s and a 3 since 12 = 2*2*3).
If j is divisible by 10, it means j has 10 as a factor too (so j has at least one 2 - we already know that from above - and a 5).

So we know that j has at least two 2s, one 3 and one 5. To make 24, we need three 2s and one 3 (because 24 = 2*2*2*3). Do we know whether j has three 2s? No we don't.
j could be 2*2*2*3*5*7
it could be 2*2*3*5*7*11
or it could be 2*2*3*3*5*13
or many other things.
All we know is that it must have two 2s and one 3 and one 5. But this is not sufficient information.
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Re: Division Question Driving Me Crazy [#permalink] New post 28 Apr 2013, 01:29
You guys are seriously awesome! Sorry for the superficial feedback, I am just reading the feedback before going to bed and had to thank you guys real quick.
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Re: Division Question Driving Me Crazy [#permalink] New post 29 Apr 2013, 09:57
Thank you everybody for your help!

I found the below explanation to be absolutely illuminating! I'm really really thankful. I understand it now.

VeritasPrepKarishma wrote:
tmipanthers wrote:
Hi guys: This question is from the Number Properties book (5th ed) from Manhattan Guide. My answer is different from that stated in the solutions, so I am wondering if the book is incorrect.

So here is the question. I greatly appreciate the help as I'm about to smash my head into the wall. Thanks!

For questions #1-6, answer each question YES, NO, or CANNOT BE DETERMINED. If your answer
is CANNOT BE DETERMINED, use two numerical examples to show how the problem could
go either way. All variables in problems #1-6 are assumed to be integers.

4. If j is divisible by 12 and 10, is j divisible by 24?


Keep in mind that when you say a is divisible by b, it means 'b' is a factor of 'a'.

i.e. a = n*b
If b = 12, a can be 12/24/36/48/60.... etc

If j is divisible by 12, it means j has 12 as a factor (so j has at least two 2s and a 3 since 12 = 2*2*3).
If j is divisible by 10, it means j has 10 as a factor too (so j has at least one 2 - we already know that from above - and a 5).

So we know that j has at least two 2s, one 3 and one 5. To make 24, we need three 2s and one 3 (because 24 = 2*2*2*3). Do we know whether j has three 2s? No we don't.
j could be 2*2*2*3*5*7
it could be 2*2*3*5*7*11
or it could be 2*2*3*3*5*13
or many other things.
All we know is that it must have two 2s and one 3 and one 5. But this is not sufficient information.
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Divisible question for the experts [#permalink] New post 23 Jul 2013, 18:17
Hey guys,

Just starting to get into quant studies and I came across a stumper. The question states, "if j is divisible by 12 and 10, is j divisible by 24"

I created prime boxes for 12 and 10 and deduced it had to be since 3x2x2x2 = 24.

However my Manhattan GMAT book states it cannot be determined since there are only two 2's that are definitely in the prime factorization of j because the 2 in the prime factorization of 10 may be redundant as one of the 2's in the prime factorization of 12.

I have to admit, I don't follow that at all. Any assistance would be greatly appreciated.

Thanks!

I should note, I can't believe how much math I have forgotten. I need the basic refresher imaginable...
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Re: Divisible question for the experts [#permalink] New post 23 Jul 2013, 21:35
Expert's post
TheLostOne wrote:
Hey guys,

Just starting to get into quant studies and I came across a stumper. The question states, "if j is divisible by 12 and 10, is j divisible by 24"

I created prime boxes for 12 and 10 and deduced it had to be since 3x2x2x2 = 24.

However my Manhattan GMAT book states it cannot be determined since there are only two 2's that are definitely in the prime factorization of j because the 2 in the prime factorization of 10 may be redundant as one of the 2's in the prime factorization of 12.

I have to admit, I don't follow that at all. Any assistance would be greatly appreciated.

Thanks!

I should note, I can't believe how much math I have forgotten. I need the basic refresher imaginable...


Merging similar topics. Please go through the posts above and ask if anything remains unclear.

Generally if positive integer j is divisible by 12 and 10, then it must be divisible by the least common multiple of 12 and 10, so by 60. This implies that j can be 60, 120, 180, ... Now, if j=60, then it won't be divisible by 24 but if it's for example, 120 then it will be.

The following DS question might also help: is-j-divisible-by-138043.html
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Re: False proof way to solve number property questions like [#permalink] New post 23 Jul 2013, 23:42
anuj4ufriends wrote:
What is the false proof way of solving questions like, 'If j is divisible by 12 and 10, is j divisible by 24?'
(a) Yes
(b) No
(c) Cannot be determined

How I happened to solve this question is:
2,2,3 are factors of 12.
2,5 are factors of 10.

Since 24 is 8*3 which can be formed from above factors of 12 and 10(as per factor foundation rule), j is divisible by 24.

But, the answer is 'Cannot be determined', since he rightly shows that if j=60 then 60 is not divisible by 24, where as it is divisible by 12, 10.
Thus, can anyone show me a false proof procedure(without falling into any traps as above), to solve the above type of questions?


j is divisible by 12 and 10. So taking LCM of both these numbers, we can say that j is divisible by 60.

j/60
Now, we can say j=60k, where k can be 1,2,3,4...

j/24=> 60k/24 => 5k/2

if k is even, j/24 is valid else invalid... So, we cannot determine...
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Re: False proof way to solve number property questions like   [#permalink] 23 Jul 2013, 23:42
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