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

Re: If r and s are positive integers, is r/s an integer? [#permalink]
11 Apr 2012, 12:25

Bunuel what about the word "every" - according to this i picked up numbers which satisfied this condition. And one more point-official explanation "takes" the numbers 18 and 8, to explain the answer. But 18 has the prime factors 2 and 3, but 8 hasn't 3 as a prime factor - what means not every factor of x is also a prime factor of y - and this contradicts with the statement I just need to clarify -to avoid such a confusion on the exam

Re: If r and s are positive integers, is r/s an integer? [#permalink]
11 Apr 2012, 12:28

Expert's post

Galiya wrote:

Bunuel what about the word "every" - according to this i picked up numbers which satisfied this condition. And one more point-official explanation "takes" the numbers 18 and 8, to explain the answer. But 18 has the prime factors 2 and 3, but 8 hasn't 3 as a prime factor - what means not every factor of x is also a prime factor of y - and this contradicts with the statement I just need to clarify -to avoid such a confusion on the exam

Let's refer to the question in the initial post: which statement are you talking about? _________________

Re: If r and s are positive integers, is r/s an integer? [#permalink]
11 Apr 2012, 12:40

Expert's post

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. _________________

Re: If r and s are positive integers, is r/s an integer? [#permalink]
24 Aug 2012, 01:04

Expert's post

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.

Re: If a and b are positive integers, is ‘a’ a multiple of b? [#permalink]
25 Sep 2012, 10:24

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. _________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

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

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

Re: If r and s are positive integers, is r/s an integer? [#permalink]
05 May 2014, 01:40

Expert's post

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).

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