DS-Number Properties : Need simple explanation to solve

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

(1) x is odd

(2) x leaves a remainder when divided by 3

OA:C

Question: Not able to understand on how to solve and need explanation for the same.
Experts please help.

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by Ian Stewart » Tue Aug 20, 2019 5:38 am
A number will be divisible by 24 if it is divisible by 3 and by 8. We want to know if x^2 - 1 = (x-1)(x+1) is divisible by 24. Notice that x-1 and x+1 are separated by 2, so they are either consecutive odd integers or consecutive even integers.

From Statement 1, we learn that x is odd, so x-1 and x+1 must be consecutive even integers. If you look at any two consecutive even integers, one of them will be divisible by 2, and the other will be divisible by 4 (at least), so the product of two consecutive even integers is always divisible by 8. So Statement 1 ensures that (x-1)(x+1) is divisible by 8, but we don't know if it's divisible by 3, and Statement 1 is not sufficient.

From Statement 2, we learn that x is not divisible by 3. But x-1, x and x+1 are three consecutive integers, and among any three consecutive integers, you always find exactly one multiple of 3, since multiples of 3 are three apart. If x is not a multiple of 3, then either x-1 or x+1 must be. So Statement 2 ensures that (x-1)(x+1) is divisible by 3, but we don't know if it's divisible by 8, and Statement 2 is not sufficient.

Using both Statements, we know (x-1)(x+1) = x^2 - 1 is divisible by 8 from Statement 1, and by 3 from Statement 2, and is therefore divisible by 24, so the answer is C.
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