OG 11th edition, practice question #125

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.

Thanks!

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?

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 ..

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

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.

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" ..

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..

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 .. 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 ..

.. anyway as u mentioned in ur post the answer A still holds ..

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 .. 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 ..

.. anyway as u mentioned in ur post the answer A still holds ..

Good discussion guys... thanks for the help...

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