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If x is a positive integer, is x^2 - 1 divisible by 24?

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axezcole
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If x is a positive integer, is x^2 - 1 divisible by 24? [#permalink] New post   Updated on: 25 Apr 2019, 01:45
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If x is a positive integer, is x^2 - 1 divisible by 24?

(1) x is odd
(2) x leaves a remainder when divided by 3
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Official Answer
C

Originally posted by axezcole on 22 Apr 2019, 23:46.
Last edited by Bunuel on 25 Apr 2019, 01:45, edited 3 times in total.
Renamed the topic and edited the question.
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LevanKhukhunashvili
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Re: If x is a positive integer, is x^2 - 1 divisible by 24? [#permalink] New post   23 Apr 2019, 03:41
Dont' agree with OA

We are asked if (x-1)(x+1) is divisible by 3*\(2^3\)
2 stm --> x=2 leaves a reminder when divided by 3 and (2-1)(2+1) is not divisible by 24; x=5 (5-1)(5+1) is divisible by 24

Imo answer is C
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nick1816
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Re: If x is a positive integer, is x^2 - 1 divisible by 24? [#permalink] New post   23 Apr 2019, 03:58
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Wrong OA
when x=4, x^2-1=15( not divisible by 24)
when x=7, x^2-1=48( divisible by 24)
Hence B is insufficient
Answer must be C
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Re: If x is a positive integer, is x^2 - 1 divisible by 24? [#permalink] New post   23 Apr 2019, 04:25
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If x is a positive integer,is \(x^2\)-1 divisible by 24?
1.x is odd
If x is odd, then x-1 and x+1 will be divisible by atleast a 2 and a 4. so \(x^2-1\), which is (x-1)(x+1) will be divisible by 2*4 or 8 at least.
Now, if any one of x-1 or x+1 is divisible by 3, answer is yes, x^2-1 is divisible by 8*3. But, if x is divisible by 3, answer is no.

2.x leaves a remainder when divided by 3
Now \(x^2-1=(x-1)(x+1)\), so if x is not divisible by 3, either x-1 or x+1 will be divisible by 3. Therefore, if x is even, the answer will be no, as both x-1 and x+1 will be odd. for example, 2, 4, or 8. Where as if x is odd, for example 7, 11 etc, answer is yes


Combined..
Statement I tells us that \(x^2\)-1 divisible by at least 8, and Statement II tells us that \(x^2\)-1 divisible by at least a 3, so \(x^2\)-1 is surely divisible by at least 8*3 or 24.
Suff

C
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Re: If x is a positive integer, is x^2 - 1 divisible by 24? [#permalink] New post   23 Apr 2019, 06:58
Apologies ,marked Option B by mistake.The correct answer is C.
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Re: If x is a positive integer, is x^2 - 1 divisible by 24? [#permalink] New post   04 May 2019, 01:29
axezcole wrote:
If x is a positive integer, is x^2 - 1 divisible by 24?

(1) x is odd
(2) x leaves a remainder when divided by 3


#1
x=1,3,5 we get variation in answers, insufficient
#2
x=4,5,7
again in sufficient
from1 & 2
x=5 issufficient
IMO C
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Re: If x is a positive integer, is x^2 - 1 divisible by 24? [#permalink] New post   11 Jun 2020, 04:26
axezcole wrote:
If x is a positive integer, is x^2 - 1 divisible by 24?

(1) x is odd
(2) x leaves a remainder when divided by 3


x≥1 (positive integer)
x^2-1=(x+1)(x-1)
factors(24)=2^3*3

(1) insufic
x=1: 0(2)/24=div
x=3: 2(4)/24≠div

(2) insufic
x=1: 0(2)/24=div
x=2: 1(3)/24≠div

(1/2) sufic
x=1: 0(2)/24=div
x=5: 4(6)/24=div
x=7: 6(8)/24=div

Ans (C)
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Re: If x is a positive integer, is x^2 - 1 divisible by 24? [#permalink]
11 Jun 2020, 04:26
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Bunuel
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