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x is positive integer, is (x-1)*(x +1). divisible by 24?

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x is positive integer, is (x-1)*(x +1). divisible by 24? [#permalink] New post 09 Feb 2012, 18:03
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x is positive integer, is (x-1)*(x +1). divisible by 24?

(1) x is not divisible by 3
(2) x-1 is divisible by 2
[Reveal] Spoiler: OA

Last edited by Bunuel on 09 Feb 2012, 23:46, edited 1 time in total.
Edited the question and added the OA
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x is positive integer, is (x-1)*(x +1). divisible by 24? [#permalink] New post 09 Feb 2012, 23:45
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x is positive integer, is (x-1)*(x +1). divisible by 24?

(1) x is not divisible by 3 --> since (x-1), x and (x +1) are 3 consecutive integers then out of them one must be divisible by 3, as its not x then it must be either (x-1) or (x +1). So we know that (x-1)(x +1) is divisible by 3. Still not sufficient, for example if x=2 then the answer would be NO but if x=1 then the answer would be YES.

(2) x-1 is divisible by 2 --> (x-1) is even --> (x-1) and (x +1) are consecutive even integers, thus one of them is divisible by 4 --> (x-1)*(x +1) is divisible by 2*4=8. Still not sufficient, for example if x=3 then the answer would be NO but if x=1 then the answer would be YES.

(1)+(2) From (1) (x-1)(x +1) is divisible by 3 and from (2) its divisible by 8, hence it's divisible by 3*8=24 (the LCM of 3 and 8). Sufficient.

Answer: C.
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Re: x is positive integer, is (x-1)*(x +1). divisible by 24? [#permalink] New post 30 Dec 2014, 03:28
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Re: x is positive integer, is (x-1)*(x +1). divisible by 24? [#permalink] New post 31 Dec 2014, 12:03
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Hi All,

This question can be solved by TESTing VALUES, although you'll have to do a little bit of extra work at the end to prove that your answer is correct.

We're told that X is a POSITIVE INTEGER. We're asked if (X-1)(X+1) is divisible by 24. This is a YES/NO question.

Fact 1: X is NOT divisible by 3.

IF...
X = 1
(0)(2) = 0 and the answer to the question is YES.

X = 2
(1)(3) = 3 and the answer to the question is NO.
Fact 1 is INSUFFICIENT

Fact 2: X-1 is divisible by 2

This means that X MUST be ODD.

IF...
X = 1, then the answer to the question is YES (we did the work in Fact 1 already).

X = 3
(2)(4) = 8 and the answer to the question is NO
Fact 2 is INSUFFICIENT

Combined, we know...
X MUST be odd
X is NOT divisible by 3

IF....
X = 1, then the answer to the question is YES

X = 5
(4)(6) = 24 and the answer to the question is YES

X = 7
(6)(8) = 48 and the answer to the question is YES

X = 11
(10)(12) = 120 and the answer to the question is YES

We have a consistent result.
Combined, SUFFICIENT

Final Answer:
[Reveal] Spoiler:
C


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Re: x is positive integer, is (x-1)*(x +1). divisible by 24?   [#permalink] 31 Dec 2014, 12:03
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