If r and s are positive integers, is r/s an integer? : GMAT Data Sufficiency (DS)
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# If r and s are positive integers, is r/s an integer?

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

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23 Feb 2012, 07:13
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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.
[Reveal] Spoiler: OA
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Re: If r and s are positive integers, is r/s an integer? [#permalink]

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23 Feb 2012, 07:26
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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.

Answer: A.

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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26 Oct 2015, 02:37
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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

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

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24 Jun 2015, 03:44
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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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11 Apr 2012, 12:30
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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11 Apr 2012, 12:40
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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23 Aug 2012, 20:14
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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24 Aug 2012, 01:04
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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25 Sep 2012, 10:24
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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.

Answer B.
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DS : X and Y are two positive integers [#permalink]

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24 Oct 2012, 22:25
Hi,

I'm trying to figure out this one.

X and Y are two positive integers. Is x divided by y an integer?

(1) All of Y's factors are also factors of X.

(2) Each prime factor of Y is also a prime factor of X.

Any pointers, please?

A, B, C, D or E?

Thanks.
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Re: DS : X and Y are two positive integers [#permalink]

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24 Oct 2012, 22:33
elegan wrote:
Hi,

I'm trying to figure out this one.

X and Y are two positive integers. Is x divided by y an integer?

(1) All of Y's factors are also factors of X.

(2) Each prime factor of Y is also a prime factor of X.

Any pointers, please?

A, B, C, D or E?

Thanks.

1) Y itself is a factor of Y. Hence Y is also a factor of X. Sufficient

2)Suppose X = 2 and Y = 8. Answer is No. If Y = 2 and X =8 answer is yes. Insufficient

Answer is A.

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

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12 Jan 2014, 05:35
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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04 May 2014, 10:23
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.

Answer: A.

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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05 May 2014, 01:40
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.

Answer: A.

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

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26 Jul 2014, 10:21
Factors of 4 is 1, 2 and 4. Factors of 6 is 1, 2, 3 and 6. Therefore, your example is not a true representation of statement 1. If every factor of s is also a factor of r, then what it is really saying is that r is a multiple of s, i.e. Factor of $$s=4$$ is 1, 2 and 4. Factor of $$r=8$$ is 1, 2, 4 and 8, while $$r=12$$ has factors 1, 2, 3, 4, 6 and 12. Since multiples are defined as whole numbers, it is sufficient.

Statement 2 is not sufficient since it is simply stating that each prime factor of s is also a prime factor of r. It does not state how many of each prime factor is present.
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Re: If r and s are positive integers, is r/s an integer? [#permalink]

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24 Jun 2015, 03:40
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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If r and s are positive integers, is r/s an integer? [#permalink]

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27 Mar 2016, 18:06
For condition #1, remember that this is just a fancy way of indicating that r > s.
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Re: If r and s are positive integers, is r/s an integer? [#permalink]

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31 Mar 2016, 00:02
Attached is a visual that should help.
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Screen Shot 2016-03-31 at 1.01.50 AM.png [ 145.85 KiB | Viewed 5384 times ]

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

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15 Sep 2016, 06:06
My suggestion: combine set properties with number properties. It becomes more intelligible.
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Re: If r and s are positive integers, is r/s an integer?   [#permalink] 15 Sep 2016, 06:06
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