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OG 11th edition, practice question #125

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by naren_nayak » Fri Jul 06, 2007 9:45 am
If r & s are positive integers, is r/s an integer?

(1) Every factor of s is also a factor of r
(2) Every prime factor of s is also a prime factor of r

OA is A. I don't get it. Isn't the following possible?

Scenario 1: r = 4 = 1*2*2, s = 2 = 1*2; r/s = 4/2 => r/s is an integer

Scenario 2: r = 5 = 1*2*2.5 , s = 2 = 1*2; r/s = 1*2*2.5 / 1*2 => r/s is not an integer

Thanks in advance!
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by givemeanid » Fri Jul 06, 2007 10:09 am
If r = 5, 2.5 is not a 'factor' of r. Factor of an integer n is an integer that divides n without leaving a remainder. 2.5 is not an integer and hence not a factor of 5.
So It Goes
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by naren_nayak » Fri Jul 06, 2007 12:37 pm
Thanks!
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Re: OG 11th edition, practice question #125

by bingojohn » Wed Aug 08, 2007 7:50 am
naren_nayak wrote:If r & s are positive integers, is r/s an integer?

(1) Every factor of s is also a factor of r
(2) Every prime factor of s is also a prime factor of r

OA is A. I don't get it. Isn't the following possible?

Scenario 1: r = 4 = 1*2*2, s = 2 = 1*2; r/s = 4/2 => r/s is an integer

Scenario 2: r = 5 = 1*2*2.5 , s = 2 = 1*2; r/s = 1*2*2.5 / 1*2 => r/s is not an integer

Thanks in advance!
I think the answer is [D], each alone. I don't get how the answer should be [A].

if I prime factorize r and s, r has all the prime factors of s and then some more prime factors... so, we can easily say

r = s (k), where k is a positive integer equal to the product of the excess prime factors of r (in excess of those of s).

which means r/s is an integer

Anything wrong with that?
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Re: OG 11th edition, practice question #125

by gabriel » Wed Aug 08, 2007 8:08 am
bingojohn wrote:
naren_nayak wrote:If r & s are positive integers, is r/s an integer?

(1) Every factor of s is also a factor of r
(2) Every prime factor of s is also a prime factor of r

OA is A. I don't get it. Isn't the following possible?

Scenario 1: r = 4 = 1*2*2, s = 2 = 1*2; r/s = 4/2 => r/s is an integer

Scenario 2: r = 5 = 1*2*2.5 , s = 2 = 1*2; r/s = 1*2*2.5 / 1*2 => r/s is not an integer

Thanks in advance!
I think the answer is [D], each alone. I don't get how the answer should be [A].

if I prime factorize r and s, r has all the prime factors of s and then some more prime factors... so, we can easily say

r = s (k), where k is a positive integer equal to the product of the excess prime factors of r (in excess of those of s).

which means r/s is an integer

Anything wrong with that?

ok the second statement says ,

(2) Every prime factor of s is also a prime factor of r ..

the statement talks about prime factors ..

Consider 2 numbers 12,6 .. 12 = 2^2*3 and 6 = 2*3 .. over here the prime factors for both 12 and 6 are 2 and 3 .. so r could be either 12 or 6 .. so, r/s may or may not be a integer .. hope that helps ..
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Re: OG 11th edition, practice question #125

by bingojohn » Wed Aug 08, 2007 1:42 pm
gabriel wrote:
bingojohn wrote:
naren_nayak wrote:If r & s are positive integers, is r/s an integer?

(1) Every factor of s is also a factor of r
(2) Every prime factor of s is also a prime factor of r

OA is A. I don't get it. Isn't the following possible?

Scenario 1: r = 4 = 1*2*2, s = 2 = 1*2; r/s = 4/2 => r/s is an integer

Scenario 2: r = 5 = 1*2*2.5 , s = 2 = 1*2; r/s = 1*2*2.5 / 1*2 => r/s is not an integer

Thanks in advance!
I think the answer is [D], each alone. I don't get how the answer should be [A].

if I prime factorize r and s, r has all the prime factors of s and then some more prime factors... so, we can easily say

r = s (k), where k is a positive integer equal to the product of the excess prime factors of r (in excess of those of s).

which means r/s is an integer

Anything wrong with that?

ok the second statement says ,

(2) Every prime factor of s is also a prime factor of r ..

the statement talks about prime factors ..

Consider 2 numbers 12,6 .. 12 = 2^2*3 and 6 = 2*3 .. over here the prime factors for both 12 and 6 are 2 and 3 .. so r could be either 12 or 6 .. so, r/s may or may not be a integer .. hope that helps ..
Not clear enough, Gabriel...

Are you saying that out of the two numbers, r could be the smaller one and so r/s may not end up as an integer? If yes, isn't that true for statement 1 also? Is your answer [E], then? I still don't get how it would be [A]... basically I am not seeing the difference between statement 1 and 2 that makes statement 1 work and not 2 ... any help is appreciated
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by tanyajoseph » Thu Aug 09, 2007 12:01 am
STmt 1 says - every factor of s is also a factor of r
which means every factor of s has a clone and so at the least r/s has to be 1 or greater which makes it an integer. SO stmt 1 should be enough.
The clue here is EVERY FACTOR OF S

Hope that helps.
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Re: OG 11th edition, practice question #125

by gabriel » Thu Aug 09, 2007 1:03 am
bingojohn wrote:
Not clear enough, Gabriel...

Are you saying that out of the two numbers, r could be the smaller one and so r/s may not end up as an integer? If yes, isn't that true for statement 1 also? Is your answer [E], then? I still don't get how it would be [A]... basically I am not seeing the difference between statement 1 and 2 that makes statement 1 work and not 2 ... any help is appreciated
.. ok let me try again ..

heres the question ..

If r & s are positive integers, is r/s an integer?

(1) Every factor of s is also a factor of r
(2) Every prime factor of s is also a prime factor of r

.. now, for r/s to be a integer .. there are 2 ways this can be done .. first r and s are the same number .. then r/s =1 .. and r is a multiple of s that is r = k*s .. where k is an integer .. so in this case r/s = k ..

now, look at the first statement .. it says that every factor of is also a factor of r .. what does that mean ? .. it means that either they are one and the same number .. that is r = s .. so r/s = 1 .. or it means that r is a multiple of s .. that is if s = 6 .. then r = 6,12,24 etc ... so in this case too r/s is an integer ..

the second statement says .. every prime factor of s is also a factor of r .. the second statement talks only of the prime factors so if s = 12 with the prime factors as 2 and 3 .. then r can be 6,12,24 etc .. each of which has a prime factor of 2 and 3 .. so in this case r/s may or may not be an integer .. so the answer is A ..

the key words over here ( as tanya has mentioned in her post) are " every factor " and "every prime factor" ..
Last edited by gabriel on Thu Aug 09, 2007 6:45 am, edited 1 time in total.
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by UmanG » Thu Aug 09, 2007 3:07 am
Hi Gabrial,

Good reasoing and agreeing with you. But just want to correct you in following statement,

now, look at the first statement .. it says that every factor of is also a factor of r .. what does that mean ? .. it simply means they are one and the same number .. that is r = s .. so r/s = 1 ..

Let say r =12 = 2*2*3 and s = 6 = 2*3

Here still statement given in the question "every factor of s also a factor of r" holds and still both the number are not same. Anyways r/s will be integer only and answer remains A only. However, my point is it's not necessary that r/s will be 1 always.

Just to avoid any further confusion..:)
Thanks,
UmanG - restless mind..
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by gabriel » Thu Aug 09, 2007 6:42 am
UmanG wrote:
.. Let say r =12 = 2*2*3 and s = 6 = 2*3

Here still statement given in the question "every factor of s also a factor of r" holds and still both the number are not same...



OMG :shock: .. i cant believe that i just did that .. ur right .. i dont know what i was thinking when i wrote that post ... bcoz i myself dont agree with that post :wink: ..

.. anyway as u mentioned in ur post the answer A still holds ..
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by bingojohn » Thu Aug 09, 2007 7:51 am
gabriel wrote:
UmanG wrote:
.. Let say r =12 = 2*2*3 and s = 6 = 2*3

Here still statement given in the question "every factor of s also a factor of r" holds and still both the number are not same...



OMG :shock: .. i cant believe that i just did that .. ur right .. i dont know what i was thinking when i wrote that post ... bcoz i myself dont agree with that post :wink: ..

.. anyway as u mentioned in ur post the answer A still holds ..
Good discussion guys... thanks for the help...
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by Kansonne » Fri Dec 28, 2007 4:19 pm
Hi---

Was anyone else confused like me on the wording of this problem?

I had no issues with (1) being sufficient. I had originally thought D was the answer. For me the confusion was in the differences between (1) and (2) as to what counts as "every" whatever factor.

Interpretation #1 (Correct according to the OA)-

If r = 6 and s = 12,

r= 2 * 3
____________
s= 2 * 2 * 3

s contains two DISTINCT prime factors, 2 and 3.
r contains two DISTINCT prime factors, 2 and 3.

r/s could be integer or a fraction. Insufficient.

Interpretation #2 (Not correct)-


r= 2 * 2 * 3 <----...which means that r must have those SAME factors in the SAME quantities.
____________
s= 2 * 2 * 3 <----s has two 2's and 1 three for prime factors...


Sufficient.
__________________________________________________
I guess my question is this, how are we to know when the GMAT is asking for the same factors (the number 12 has 2 and 3 for prime factors as does the number 6.) vs. the same factors in the same quantities (12 has two powers of 2 and one power of 3 but 6 only has one power of two and one power of 3)?

Thx.

K
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