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If r and s are positive integers, is r/s an integer?  [#permalink]

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Difficulty:   35% (medium)

Question Stats: 64% (01:01) correct 36% (01:14) wrong based on 785 sessions

### HideShow timer Statistics If r and 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.
Math Expert V
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Posts: 56244
Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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18
15
If r and s are positive integers, is r/s an integer?

(1) Every factor of s is also a factor of r. If every factor of s is also factor of r, then in fraction r/s, s will just be reduced and we get an integer. Sufficient.

(2) Every prime factor of s is also a prime factor of r. The powers of prime factors of s could be higher than powers of prime factors of r. eg 25/125=1/5 not an integer. Not sufficient.

Hope it's clear.
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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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i mean this one
(2) Every prime factor of s is also a prime factor of r.
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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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Galiya wrote:
i mean this one
(2) Every prime factor of s is also a prime factor of r.

There is an example given in my post which satisfies the given condition and doesn't give an integer value of r/s.

(2) Every prime factor of s is also a prime factor of r. The powers of prime factors of s could be higher than powers of prime factors of r.

For example: if s=5^3 and r=5^2 then every prime of 125 (in fact its only prime 5) IS also a prime of 25 but r/s=25/125 is not an integer.
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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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What if S>R? It doesn't say that R>S. It just says they are positive integers.

For first statement:
R>S = 20/10 = 2 --> Integer
S>R = 10/20 = .5 --> Not an integer

For the second statement:
R>S = 50/5 = 10 --> Integer
S>R = 5/50 = .5 --> Not an integer

That is why I picked E. Can you please explain why my reasoning is incorrect? Thanks!
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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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KAS1 wrote:
What if S>R? It doesn't say that R>S. It just says they are positive integers.

For first statement:
R>S = 20/10 = 2 --> Integer
S>R = 10/20 = .5 --> Not an integer

For the second statement:
R>S = 50/5 = 10 --> Integer
S>R = 5/50 = .5 --> Not an integer

That is why I picked E. Can you please explain why my reasoning is incorrect? Thanks!

For the first statement s cannot be greater than r. If every factor of s is also factor of r, then $$r\geq{s}$$. Your example, (r=10 and s=20), is not possible, because 4 is a factor of s but not a factor of r.

Hope it's clear.
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Re: If a and b are positive integers, is ‘a’ a multiple of b?  [#permalink]

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ankit0411 wrote:
If a and b are positive integers, is ‘a’ a multiple of b?

(1) Every prime factor of b is also a prime factor of a (2) Every factor of b is also a factor of a

(1) Consider for example $$a = 2\cdot3=6$$ and $$b = 2^2\cdot3=12$$, so obviously $$a$$ is not a multiple of $$b.$$
If $$a = 12$$ and $$b = 6$$, then of course $$a$$ is a multiple of $$b.$$
Not sufficient.

(2) $$b$$ is a factor of itself, so it is also a factor of $$a$$, which means that $$a$$ is a multiple of $$b.$$
Sufficient.

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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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BANON wrote:
If r and 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.

Basically, the question asks us if s is a factor of r.

1) This is significant, because we are told something about every factor of s. Let's say the products of the factors of s = n, then 1) gives us n*r.. is n*r/n an integer? Of course.. So 1) is sufficient

2) This only tells us a fraction of the information that 1) tells us, since 2) restricts the factors to primes.. But we don't know if s has other factors that are NOT shared by r, and thus 2) is insufficient..

So A is correct
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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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Bunuel wrote:
If r and s are positive integers, is r/s an integer?

(1) Every factor of s is also a factor of r. If every factor of s is also factor of r, then in fraction r/s, s will just be reduced and we get an integer. Sufficient.

(2) Every prime factor of s is also a prime factor of r. The powers of prime factors of s could be higher than powers of prime factors of r. eg 25/125=1/5 not an integer. Not sufficient.

Hope it's clear.

Hi Bunuel,

If we look at the two statements above:
(1) Every factor of s is also a factor of r.
(2) Every prime factor of s is also a prime factor of r.

I'm having a hard time differentiating the two statements. I realize that one is talking about PF and one is talking about All Factors, but how can we assume that in statement 2, S could have it's factors raised to a higher value. Isn't the verbiage between 1 and 2 identical?
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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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russ9 wrote:
Bunuel wrote:
If r and s are positive integers, is r/s an integer?

(1) Every factor of s is also a factor of r. If every factor of s is also factor of r, then in fraction r/s, s will just be reduced and we get an integer. Sufficient.

(2) Every prime factor of s is also a prime factor of r. The powers of prime factors of s could be higher than powers of prime factors of r. eg 25/125=1/5 not an integer. Not sufficient.

Hope it's clear.

Hi Bunuel,

If we look at the two statements above:
(1) Every factor of s is also a factor of r.
(2) Every prime factor of s is also a prime factor of r.

I'm having a hard time differentiating the two statements. I realize that one is talking about PF and one is talking about All Factors, but how can we assume that in statement 2, S could have it's factors raised to a higher value. Isn't the verbiage between 1 and 2 identical?

No, they are not identical.

(2) says that r and s have the same primes but this does not mean that r and s have the same factors. For, example, 2, 4, 8, 16, ..., 2^n all have the same prime: 2. But they certainly do not share all their factors: the factors of 8 (1, 2, 4, 8) are not the same as the factors of 4 (1, 2, 4).

Does this make sense?
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GMAT 1: 730 Q49 V40 Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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Hi, I answered E to this question because I thought that "every factor of s is also factor of r" does not mean that every factor of r is also a factor of s. and that is r (6) and s (4) could be an answer: every factor of s (1 and 2) are factors of r but not all factors of r are necessarily factors of s. what am i missreading? thanks!!
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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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petu wrote:
Hi, I answered E to this question because I thought that "every factor of s is also factor of r" does not mean that every factor of r is also a factor of s. and that is r (6) and s (4) could be an answer: every factor of s (1 and 2) are factors of r but not all factors of r are necessarily factors of s. what am i missreading? thanks!!

If s = 4 and r = 6, then 4, a factor of s = 4, is NOT a factor of r = 6.
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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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4
r and s are positive integers.

$$\frac{r}{s}$$ is an integer?

Statement 1

Every factor of s is also a factor of r

Hence sufficient

Statement 2

every prime factor of s is also a prime factor of r

case 1
r = 10 therefore prime factors 2 * 5
s = 10 therefore prime factors 2 * 5

$$\frac{r}{s} = \frac{10}{10}= integer$$

case 2
r = 10 therefore prime factors 2 * 5
s= 20 therefore prime factors 2 * 5

$$\frac{r}{s} = \frac{10}{20}\neq{integer}$$

hence not sufficient

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If r and s are positive integers, is r/s an integer?  [#permalink]

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For condition #1, remember that this is just a fancy way of indicating that r > s.

Attached is a visual that should help.
Attachments Screen Shot 2016-03-31 at 1.01.50 AM.png [ 145.85 KiB | Viewed 25527 times ]

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GMAT 1: 470 Q30 V20 GMAT 2: 620 Q42 V33 Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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My suggestion: combine set properties with number properties. It becomes more intelligible. Attachments Screenshot_20160915-110321.jpg [ 1.04 MiB | Viewed 22569 times ]

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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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Bunuel wrote:
If r and s are positive integers, is r/s an integer?

(1) Every factor of s is also a factor of r. If every factor of s is also factor of r, then in fraction r/s, s will just be reduced and we get an integer. Sufficient.

(2) Every prime factor of s is also a prime factor of r. The powers of prime factors of s could be higher than powers of prime factors of r. eg 25/125=1/5 not an integer. Not sufficient.

Hope it's clear.

For statement 1, how do we know that we won't run into a similar problem with statement two in which we could have 2^2*3^2/2^3*3^3?
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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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Bunuel wrote:
If r and s are positive integers, is r/s an integer?

(1) Every factor of s is also a factor of r. If every factor of s is also factor of r, then in fraction r/s, s will just be reduced and we get an integer. Sufficient.

(2) Every prime factor of s is also a prime factor of r. The powers of prime factors of s could be higher than powers of prime factors of r. eg 25/125=1/5 not an integer. Not sufficient.

Hope it's clear.

For statement 1, how do we know that we won't run into a similar problem with statement two in which we could have 2^2*3^2/2^3*3^3?

Hello

We CANNOT take your example, where r = 2^2 * 3^2 and s = 2^3 * 3^3, as this will violate the condition in statement 1.

Statement 1 says that every factor of s is also a factor of r. But if s has more powers of 2 and/or 3 than r, then s will have some factors which are not factors of r. Eg, in your example, 2^3, 3^3, 2^3 * 3^2 .. etc.. are factors of s but they are not factors of r.

So if we go by statement 1 condition, if all factors of s have to be factors of r also, then all powers of all prime numbers contained in s, will also be contained in r. Hence r/s has to be an integer.
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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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In 1st case If s=8 and r=12 , then r/s doesn't give an integer. In this case A is not sufficient.
Am I right?
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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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sarvesh93sah wrote:
In 1st case If s=8 and r=12 , then r/s doesn't give an integer. In this case A is not sufficient.
Am I right?

If you were right then the OA would not be A. So, there must be a mistake in your reasoning. s = 8 and r = 12 does not satisfy the first statement which says: EVERY factor of s is also a factor of r. One of the factors of s = 8, is 8 itself (recall that an integer is a factor of itself) and 8 is not a factor of r = 12.
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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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BANON wrote:
If r and 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.

it matters which one is greater. if s is greater than r, it's impossible for r/s to be a whole number (it will automatically be a fraction less than 1).

first, rephrase the question: is s a factor of r? or, is r a multiple of s? or, is r divisible by s? etc.
you shouldn't leave the question in its original form; the original form is just sort of ugly.

(1)
here's the easiest way to handle this one:
s is a factor of itself.
therefore, since every factor of s (including s itself) is a factor of r, s is a factor of r.
sufficient.

you can do this part with a whole lot of song and dance involving prime numbers, but i like this argument because it's simple and elegant.

(2)
this just means that r and s are combinations of the same prime factors, but you don't know how many times each of those prime factors appears. for instance, 10 and 1000 are both made up of 2's and 5's. therefore, (r, s) could be (10, 1000), in which case the answer is 'no', or (1000, 10), in which case the answer is 'yes'.
insufficient.

ans = a

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_________________ Re: If r and s are positive integers, is r/s an integer?   [#permalink] 29 Oct 2018, 12:39
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