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# If r and s positif integers, is r/s an interger ? 1 - every

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If r and s positif integers, is r/s an interger ? 1 - every [#permalink]  09 Sep 2005, 05:54
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If r and s positif integers, is r/s an interger ?

1 - every factor of s is a factor of r
2 - every prime factor of s is also a prime factor of r
Manager
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I would say D.

1) if eachfactor of S is a factor of R==> S= 21 (21,7,3,1) R= an integer that has all the factors of S) R=63

R:S= integer

Sufficent

2) Every prime factor of S is a prime factor of R:

Sufficent as R:S will allways be a multiple of 2 ...thus an integer
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I wud say A...

(1) is obvious..

(2) lets say r=6 and s=12

then the prime factors of 6 are 2,3 and the prime factors of 12 are 2,3

6/12 is not a integer

however if r=12, s=6....then obviously r/s is an integer...

Note: the question stem is not very clear, it should say distinct prime factors...which is what I would take to be the case here...
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fresinha12 wrote:
I wud say A...

(1) is obvious..

(2) lets say r=6 and s=12

then the prime factors of 6 are 2,3 and the prime factors of 12 are 2,3

6/12 is not a integer

however if r=12, s=6....then obviously r/s is an integer...

Note: the question stem is not very clear, it should say distinct prime factors...which is what I would take to be the case here...

So if we assume that the stem is: prime factors instead of distinct prime factors it means that factoring 12= 2x2x3=
We would include 2 twice..right? and therefore r/s would be an integer?

Thanks
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We had this question last month.

Because s is a factor of itself, then r must be a multiple of s, hence r/s is an integer, so (1) holds.

Counter-example to (2), consider r/s = 2/4, but 2 is a factor of 4.
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(1) sufficient... should be evident but you can try plugging in any #'s you want to test it out

(2) prime factor similarities b/w r and s maybe work for some but not all... the best example in plugging in would be using r = 2 and s = 4 as they both share the prime factor of 2... r/s for this would not result in an integer --> insufficient

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