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

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

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19 Jul 2017, 19:50
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.

This question is relatively straightforward

St 1

Every factor of s in also a factor of r- this means that every factor of s must be a factor of r- so in other words s must be the same size as r -neither can be larger

St 2

2 x3 /3 x 2 x 2 this satisfies the condition but is obviously not an integer so insufficient

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

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16 Mar 2018, 07:53
sTATEMENT 1: It means r is greater than or equal to s. not r is greater than s.

mcelroytutoring wrote:
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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13 May 2018, 21:52
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.

For statement 1, how do we know that we won't run into a similar problem with statement two in which we could have 2^2*3^2/2^3*3^3?
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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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13 May 2018, 22:26
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.

For statement 1, how do we know that we won't run into a similar problem with statement two in which we could have 2^2*3^2/2^3*3^3?

Hello

We CANNOT take your example, where r = 2^2 * 3^2 and s = 2^3 * 3^3, as this will violate the condition in statement 1.

Statement 1 says that every factor of s is also a factor of r. But if s has more powers of 2 and/or 3 than r, then s will have some factors which are not factors of r. Eg, in your example, 2^3, 3^3, 2^3 * 3^2 .. etc.. are factors of s but they are not factors of r.

So if we go by statement 1 condition, if all factors of s have to be factors of r also, then all powers of all prime numbers contained in s, will also be contained in r. Hence r/s has to be 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 2018, 11:11
In 1st case If s=8 and r=12 , then r/s doesn't give an integer. In this case A is not sufficient.
Am I right?
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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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23 Aug 2018, 19:55
sarvesh93sah wrote:
In 1st case If s=8 and r=12 , then r/s doesn't give an integer. In this case A is not sufficient.
Am I right?

If you were right then the OA would not be A. So, there must be a mistake in your reasoning. s = 8 and r = 12 does not satisfy the first statement which says: EVERY factor of s is also a factor of r. One of the factors of s = 8, is 8 itself (recall that an integer is a factor of itself) and 8 is not a factor of r = 12.
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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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29 Oct 2018, 11:39
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.

it matters which one is greater. if s is greater than r, it's impossible for r/s to be a whole number (it will automatically be a fraction less than 1).

first, rephrase the question: is s a factor of r? or, is r a multiple of s? or, is r divisible by s? etc.
you shouldn't leave the question in its original form; the original form is just sort of ugly.

(1)
here's the easiest way to handle this one:
s is a factor of itself.
therefore, since every factor of s (including s itself) is a factor of r, s is a factor of r.
sufficient.

you can do this part with a whole lot of song and dance involving prime numbers, but i like this argument because it's simple and elegant.

(2)
this just means that r and s are combinations of the same prime factors, but you don't know how many times each of those prime factors appears. for instance, 10 and 1000 are both made up of 2's and 5's. therefore, (r, s) could be (10, 1000), in which case the answer is 'no', or (1000, 10), in which case the answer is 'yes'.
insufficient.

ans = a

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

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01 Nov 2018, 21:19
1. Either r is a multiple of s or equal to it. In either case, r/s is an integer. Sufficient.
2. For r=100, s=10, r/s is an integer. For r=100, s=1000, r/w is not an integer. In each case, prime factors of s are each factors of r. Insufficient.

A.
Re: If r and s are positive integers, is r/s an integer? &nbs [#permalink] 01 Nov 2018, 21:19

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