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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]  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]  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.

Hope it's clear.
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is x/y is an integer [#permalink]  11 Apr 2012, 12:05
All
could you provide ur thoughts on this please?

if x and y are positive integers, is x/y is an integer?

(1) every prime factor of x is also the prime factor of y
(2) every factor of x is also a factor of y

i went with D, since the word "every" confused me
but seems missed something
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Re: is x/y is an integer [#permalink]  11 Apr 2012, 12:11
Galiya wrote:
All
could you provide ur thoughts on this please?

if x and y are positive integers, is x/y is an integer?

(1) every prime factor of x is also the prime factor of y
(2) every factor of x is also a factor of y

i went with D, but seems missed something

if every prime factor x is also a prime factor of y - lets take an example:

X= 30 prime factors = 2, 3, 5
y = 60 prime factors = 2, 3, 5

but x/y = 1/2

whereas if all factors of x = all factors of y that simply tells us that x is y inclusive and therefore totally divisible by y.
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Re: is x/y is an integer [#permalink]  11 Apr 2012, 12:17
Expert's post
Galiya wrote:
All
could you provide ur thoughts on this please?

if x and y are positive integers, is x/y is an integer?

(1) every prime factor of x is also the prime factor of y
(2) every factor of x is also a factor of y

i went with D, since the word "every" confused me
but seems missed something

Proper version of this question is in the initial post.
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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
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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?
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Re: If r and s are positive integers, is r/s an integer? [#permalink]  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]  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.
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Re: If r and s are positive integers, is r/s an integer? [#permalink]  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]  24 Aug 2012, 01:04
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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]  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.

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

A, B, C, D or E?

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

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

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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..

So A is correct
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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.

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]  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.

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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If r and s are positive integers, is r/s an integer? [#permalink]  26 Jul 2014, 10:10
1) every factor of s is also factor of r.
2) every prime factor of s is also prime factor of r.
[Reveal] Spoiler:
Answer A) Statement 1 alone is sufficient, but 2 is not sufficient.

Why is my plugging in numbers not working here. I chose E as answer instead of A.

1) take 6 and 4 as r and s.
2 is the factor of 4 and is also factor 6. 6/4 = 3/2 , not an integer.

Does 1) mean that plug in number for r has to have both 2 and 2 as factors?
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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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Re: If r and s are positive integers, is r/s an integer? [#permalink]  26 Jul 2014, 10:37
Expert's post
GMatAspirerCA wrote:
1) every factor of s is also factor of r.
2) every prime factor of s is also prime factor of r.
[Reveal] Spoiler:
Answer A) Statement 1 alone is sufficient, but 2 is not sufficient.

Why is my plugging in numbers not working here. I chose E as answer instead of A.

1) take 6 and 4 as r and s.
2 is the factor of 4 and is also factor 6. 6/4 = 3/2 , not an integer.

Does 1) mean that plug in number for r has to have both 2 and 2 as factors?

Merging topics. Please refer to the discussion above.
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Re: If r and s are positive integers, is r/s an integer?   [#permalink] 26 Jul 2014, 10:37
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