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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, 08:13
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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, 08:26
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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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Re: If r and s are positive integers, is r/s an integer? [#permalink]

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11 Apr 2012, 13:30
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, 13:40
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, 21:14
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, 02:04
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, 11:24
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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]

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24 Oct 2012, 23:25
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]

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24 Oct 2012, 23:33
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]

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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..

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, 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.

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, 02:40
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.

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, 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.
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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, 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!!
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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, 04:44
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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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26 Oct 2015, 03:37
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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

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

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27 Mar 2016, 19:06
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, 01:02
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 7137 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, 07:06
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]

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22 May 2017, 11:33
We need to understand if r/s, if converted to decimal, would be a decimal with a finite also known as terminated decimal.

We know that the decimal number of the fraction will terminate if and ONLY in the case when the denominator of the fraction is factorized by 2 or 5 or both. If prime factorization of the denominator will include numbers other than 2 or 5, decimal of the fraction will not terminate or in other words will be indefinite.

1. s if a factor of 100 = > s must be 2, 4, 5 10, 20, 25, 50 or 100.
Factorizing 100 = 25x4 = 5^5x2^2.
100 breaks down to the prime number of only of 5 and 2 and s is a factor of 100, therefore, we know that s would contain factors of 2 and 5 with an exception of 1. But if s would have been equal to 1 => r/s = 1 as well, and that would be resulting terminal decimal. Based on this information we know that r/s is definitely terminal decimal.

Statement 1 is sufficient.

2. r is a factor of 100. => the information given is not sufficient enough to determine whether r/s decimal is terminal. The factor of 100 - 2 or 5 can be divided into numerous numbers resulting unlimited decimals.

Statement 2 is insufficient.

Re: If r and s are positive integers, is r/s an integer?   [#permalink] 22 May 2017, 11:33
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