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31 May 2023, 08:33
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Difficulty:
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Question Stats:
60% (02:47) correct
40%
(02:38)
wrong
based on 63
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If x is a positive integer greater than 1, what is the remainder when (x – 1) (x + 1) is divided by 24?
(A) 6 + x is not divisible by 3
(B) 5x is not a multiple of 2
(A) 6 + x is not divisible by 3
(B) 5x is not a multiple of 2
31 May 2023, 13:53
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ChandlerBong
Question: (x – 1) (x + 1) is divided by 24
Inference: Does (x-1)(x+1) consists of two 2s and one 3.
We can start with statement 2 as it is relatively easier between the two statements.
Statement 2
(B) 5x is not a multiple of 2
As 5x is not a multiple of 2, we can conclude that x is not even. Therefore x is odd and (x-1) and (x+1) are even.
We can be sure that (x-1) and (x+1) each consist of one 2 in it, therefore the product must at least have two 2s. However, we cannot conclude of the product has a 3 in it.
The statement alone is not sufficient. Eliminate B and D.
Statement 1
(A) 6 + x is not divisible by 3
6 is divisible by 3. Hence, as 6 + x is not divisible by 3, we can conclude that x is not divisible by 3.
Hence, x leaves a remainder of 1 or 2 when divided by three.
If x leaves a remainder of 1, (x-1) will be divisible by 3.
If x leaves a remainder of 2, (x+1) will be divisible by 3.
Therefore in all cases (x-1)(x+1) will ALWAYS be divisible by 3. While we have established the divisibility by 3, we don't know if the product has three 2s in it. Hence, the statement alone is not sufficient. We can eliminate A.
Combined
The statements combined provide us the below information
- (x-1)(x+1) is divisible by 3
- x is odd, so (x+1)(x-1) is even. As x > 1, the even number, (x+1), will have at least two 2s in it. Hence the product of (x-1)(x+1) will have at least three 2s in the product.
Therefore we can conclude that the product of (x-1)(x+1) is a multiple of \(2^3 * 3\).
The statements combined are sufficient.
Option C
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