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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.

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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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.

Re: If a and b are positive integers, is ‘a’ a multiple of b? [#permalink]

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25 Sep 2012, 11:24

1

This post was BOOKMARKED

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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PhD in Applied Mathematics Love GMAT Quant questions and running.

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

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12 Jan 2014, 06:35

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..

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

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04 May 2014, 11:23

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?

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).

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

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26 Jul 2014, 11:21

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.

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

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24 Jun 2015, 04:40

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!!

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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16 Mar 2016, 05:49

Superb Question remember here that two number may be entirely different but have the same prie factors statement 1 is sufficient as all the factors are same but two is not sufficient as PRIME factors are same hence A
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