If you were running short on time and couldn't outline the full algebraic solution, an alternate method of approaching this problem would be playing around with the prompt and testing numbers.
The question asks if x^2 - 1 is divisible by 24. If you were to test numbers, you'd find that the answer would be YES if x were 1, 5, 7, and 11. There are obviously many more values of x, but with limited time, you'd probably want to stop here.
(1) x is not divisible by 3
x = 1 and x = 2 both satisfy statement 1. The former results results in a YES answer to the prompt, while the latter results in a NO. Insufficient.
(2) x-1 is divisible by 2
That simply says x is odd. x = 1 and x = 3 both satisfy Statement 2. The former results results in a YES answer to the prompt, while the latter results in a NO. Insufficient.
(1) and (2) together - x is odd, but not divisible by 3
The fact that our YES values of x (i.e. 1, 5, 7, 11) are all odd and skip over all odd multiples of 3 is a good indication that this is very likely sufficient, so in a time-crunch situation, you would go with C.
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Now for some algebraic properties! Because you're asking about divisibility of 24, which has a prime factorization of 3 * 2^3, you need to account for 2^3 and the 3.
If (x-1)(x+1) were divisible by 24, then x-1 and x+1 would have to be consecutive even integers. One of those two even integers is necessarily a multiple of 4, meaning you have one multiple of 4 and another multiple of 2. That takes care of 2^3, but there's not necessarily a 3 there.
Next, if (x-1)(x+1) were divisible by 24, then (x-1)(x+1) would have to be divisible by 3. Therefore, either x-1 or x+1 is a multiple of 3. We've already determined that both x-1 and x+1 would have to be even, and since one has to be a multiple of 3, that also means that either x-1 or x+1 is a multiple of 6.
(1) x is not divisible by 3
x-1, x, and x+1 are consecutive integers, which means that one of them must be a multiple of 3. If x is not a multiple of 3, then it must be either x-1 or x+1. Therefore, (x-1)(x+1) is divisible by 3. But is x even? We don't know. INSUFFICIENT
(2) x-1 is divisible by 2
That means we fulfill the condition that x-1 and x+1 are both even, and thus (x-1)(x+1) is divisible by 2^3. But is it divisible by 3? We don't know. INSUFFICIENT
(1) and (2) together tell us that x is divisible by both 3 and 2^3, and thus divisible by 24. SUFFICIENT.
Final Answer: C
Rich Zwelling
GMAT Instructor, Veritas Prep
