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If x–1 is a positive integer, is x–1 divisible by 3?

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If x–1 is a positive integer, is x–1 divisible by 3? [#permalink] New post 15 May 2012, 19:06
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If x–1 is a positive integer, is x–1 divisible by 3?

(1) (x^2 + x)(x + 1) is divisible by 3.

(2) x(x^2–x) is divisible by 3.
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Re: Number Properties: Divisibility [#permalink] New post 15 May 2012, 20:32
Ljchen121 wrote:
If x– 1 is a positive integer, is x– 1 divisible by 3?

(1) (x^2 + x)(x + 1) is divisible by 3.

(2) x(x^2–x) is divisible by 3.


Hi Ljchen

use substitution for both the statements..

best Vaibhav
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Re: Number Properties: Divisibility [#permalink] New post 15 May 2012, 20:40
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narangvaibhav wrote:
Ljchen121 wrote:
If x– 1 is a positive integer, is x– 1 divisible by 3?

(1) (x^2 + x)(x + 1) is divisible by 3.

(2) x(x^2–x) is divisible by 3.


Hi Ljchen

use substitution for both the statements..

best Vaibhav


Thanks Vaibhav. This was my train of thought: just factor everything
1) x(x+1)(x+1) , either (x+1) or x must be divisible by 3, so (x-1) is not divisible by 3. SUF
2) (x)(x)(x-1), x or (x-1) must be divisible by 3, but which one, it is uncertain INS
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Re: Number Properties: Divisibility [#permalink] New post 15 May 2012, 20:52
Ljchen121,

this is about three consecutive positive (x-1 > 0 => x > 1) number x-1, x, x+1. It is evident, only one of these can be divisible by 3

So from Statements :
(1) (x^2 + x)(x + 1) is divisible by 3.
=> x (x+1) (x+1) is divisible by 3.
=> Either x OR x+1 is divisible by 3.
=> It implies (x-1) can't be divisible by 3
=> SUFFICIENT

(2) x(x^2–x) is divisible by 3.
=> x (x) (x-1) is divisible by 3.
=> Either x OR x-1 is divisible by 3.
=> It implies (x-1) may or may not be divisible by 3
=> NOT-SUFFICIENT

Answer is A
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Re: Number Properties: Divisibility [#permalink] New post 16 May 2012, 00:20
Expert's post
Ljchen121 wrote:
narangvaibhav wrote:
Ljchen121 wrote:
If x– 1 is a positive integer, is x– 1 divisible by 3?

(1) (x^2 + x)(x + 1) is divisible by 3.

(2) x(x^2–x) is divisible by 3.


Hi Ljchen

use substitution for both the statements..

best Vaibhav


Thanks Vaibhav. This was my train of thought: just factor everything
1) x(x+1)(x+1) , either (x+1) or x must be divisible by 3, so (x-1) is not divisible by 3. SUF
2) (x)(x)(x-1), x or (x-1) must be divisible by 3, but which one, it is uncertain INS


That's a correct approach.

If x–1 is a positive integer, is x–1 divisible by 3?

Notice that x–1 is an integer means that x is an integer.

(1) (x^2 + x)(x + 1) is divisible by 3 --> x(x+1)^2 is divisible by 3 --> either x or x+1 is divisible by 3. Now, out of three consecutive integers x-1, x, and x+1 ONLY ONE is divisible by 3 and since we know that it's either x or x+1 then x-1 is definitely not divisible by 3. Sufficient.

(2) x(x^2–x) is divisible by 3 --> x^2(x-1) is divisible by 3 --> either x or x-1 is divisible by 3, but we don't know which one. Not sufficient.

Answer: A.
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Re: Number Properties: Divisibility   [#permalink] 16 May 2012, 00:20
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