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.
DSNumber Properties : Need simple explanation to solve
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A number will be divisible by 24 if it is divisible by 3 and by 8. We want to know if x^2  1 = (x1)(x+1) is divisible by 24. Notice that x1 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 x1 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 (x1)(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 x1, 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 x1 or x+1 must be. So Statement 2 ensures that (x1)(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 (x1)(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.
From Statement 1, we learn that x is odd, so x1 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 (x1)(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 x1, 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 x1 or x+1 must be. So Statement 2 ensures that (x1)(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 (x1)(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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