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If x is an integer and x ≠ 31, is (31 - x)/x an integer?

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Joined: 02 Aug 2009
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If x is an integer and x ≠ 31, is (31 - x)/x an integer? [#permalink]

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10 Dec 2017, 08:25
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55% (hard)

Question Stats:

50% (01:05) correct 50% (00:46) wrong based on 59 sessions

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If x is an integer and $$x\neq{31}$$, is $$\frac{31-x}{x}$$ an integer?

(1) x > 1.
(2) x is an even number.

[Reveal] Spoiler: OA

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Re: If x is an integer and x ≠ 31, is (31 - x)/x an integer? [#permalink]

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Updated on: 11 Dec 2017, 06:33
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chetan2u wrote:
If x is an integer and $$x\neq{31}$$, is $$\frac{31-x}{x}$$ an integer?

(1) x > 1.
(2) x is an even number.

self made - on seeing a Bunuel Q

$$\frac{31-x}{x}=\frac{31}{x}-\frac{x}{x}=\frac{31}{x}-1$$

as $$x\neq{31}$$ and $$31$$ is a prime number, so the only way the above equation can be an integer is if $$x=1$$, $$-1$$ or $$-31$$

Statement 1: as $$x>1$$ so the equation is not integer. Sufficient

Statement 2: Again as $$x\neq31, 1, -1$$ or $$-31$$, so the equation will not be an integer. Sufficient

Option D

Originally posted by niks18 on 10 Dec 2017, 09:56.
Last edited by niks18 on 11 Dec 2017, 06:33, edited 2 times in total.
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Re: If x is an integer and x ≠ 31, is (31 - x)/x an integer? [#permalink]

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11 Dec 2017, 06:14
chetan2u wrote:
If x is an integer and $$x\neq{31}$$, is $$\frac{31-x}{x}$$ an integer?

(1) x > 1.
(2) x is an even number.

self made - on seeing a Bunuel Q

(31-x)/x can be written as 31/x - 1. Here -1 is already an integer, so 31/x - 1 will be an integer only when 31/x is an integer. This will be true only if x takes the following values: 1, 31, -1, -31 (its easy to see which values will satisfy because 31 is a prime number and thus has only 2 factors - 1 and 31. But the negative values of these will also result in integer). But we are already given that x is NOT equal to 31. So the only possible integer values of x which will result in 31/x being an integer are: 1, -1, -31.

(1) x > 1
This means there is simply no value x can take which will make 31/x an integer (because 1, -1, -31 are not possible here given that x > 1). So this can never be an integer. This gives us an answer as NO to the question. Sufficient.

(2) x is an even number.
Since 31 is odd, it can never be divisible by any even number. So 31/x cannot be an integer. This gives us an answer as NO to the question. Sufficient.

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Re: If x is an integer and x ≠ 31, is (31 - x)/x an integer? [#permalink]

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11 Dec 2017, 09:21
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no so much calculation required; it's simple but tricky question

(31-x)/x =>

for all x > 1 and even : (31-x) = odd so (31-x)/x [odd/even] can not be integer.
for all x > 1 and odd : (31-x) = even so (31-x)/x[even/odd] can not be integer.
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Joined: 25 Feb 2013
Posts: 1059
Location: India
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If x is an integer and x ≠ 31, is (31 - x)/x an integer? [#permalink]

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11 Dec 2017, 11:15
Sidhrt wrote:
no so much calculation required; it's simple but tricky question

(31-x)/x =>

for all x > 1 and even : (31-x) = odd so (31-x)/x [odd/even] can not be integer.
for all x > 1 and odd : (31-x) = even so (31-x)/x[even/odd] can not be integer.

Well I don't think calculation is at all required in this question even if we use any other method. the answer is visually and logically quite evident.
Nonetheless you have brought in an excellent approach.
Kudos given to you
Intern
Joined: 04 Sep 2017
Posts: 12
Re: If x is an integer and x ≠ 31, is (31 - x)/x an integer? [#permalink]

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13 Dec 2017, 07:36
Sidhrt wrote:
no so much calculation required; it's simple but tricky question

(31-x)/x =>

for all x > 1 and even : (31-x) = odd so (31-x)/x [odd/even] can not be integer.
for all x > 1 and odd : (31-x) = even so (31-x)/x[even/odd] can not be integer.

Did you mean that even/odd can not be an integer? I know the numbers for this question will be such that it will never be an integer, but it's not true in every case.
6/3, 28/7, etc are even/odd and they are integers.
Re: If x is an integer and x ≠ 31, is (31 - x)/x an integer?   [#permalink] 13 Dec 2017, 07:36
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