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

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Math Expert
Joined: 02 Sep 2009
Posts: 49300
If r and s are nonzero integers, is r/s an integer ?  [#permalink]

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10 Oct 2017, 01:16
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Difficulty:

95% (hard)

Question Stats:

40% (02:00) correct 60% (01:51) wrong based on 46 sessions

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

(1) s – 4 = (r + 2)(r – 2)
(2) r^2 – s^2 = -12

_________________
Senior Manager
Joined: 02 Jul 2017
Posts: 294
Concentration: Entrepreneurship, Technology
GMAT 1: 730 Q50 V38
If r and s are nonzero integers, is r/s an integer ?  [#permalink]

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10 Oct 2017, 01:52
1
If r and s are nonzero integers, is r/s an integer ?

(1) s – 4 = (r + 2)(r – 2)
=> $$s-4=r^2-4$$
=> $$s=r^2$$
if r =1 s=1 => $$\frac{r}{s}$$ integer True
if r = 2 s = 4 => $$\frac{r}{s}$$ not an integer
=> For $$r\neq{1}$$ => $$s>r$$ and $$\frac{r}{s}$$ is not an integer
Insufficient

(2) $$r^2 – s^2 = -12$$
=> $$s^2 – r^2 = 12$$
=> That is r and s are not equal and |s| > |r|
=> $$\frac{r}{s}$$ cannot be an integer
Sufficient

Senior Manager
Joined: 05 Dec 2016
Posts: 251
Concentration: Strategy, Finance
GMAT 1: 620 Q46 V29
Re: If r and s are nonzero integers, is r/s an integer ?  [#permalink]

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10 Oct 2017, 02:14
We need to discover whether s is a factor of r

(1)s-4=(r-2)(r+2)
s-4=r^2-4
s=r^2

s=1; r=1 => r/s=1 YES-integer
s=4; r=2 r/s= 2/4=1/2 NO

Insufficient

(2) r^2-s^2= -12
This statement tells us that |r|<|s| hence r/s would not be an integer Sufficient.

Re: If r and s are nonzero integers, is r/s an integer ? &nbs [#permalink] 10 Oct 2017, 02:14
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