kunalbh19 wrote:

If n is a positive integer, is n^2 - 1 divisible by 24?

(1) n is a prime number.

(2) n is greater than 191

We need to determine whether (n^2 - 1)/24 = integer. Notice that 24 is 2^3 x 3, or 8 x 3.

Since n^2 - 1 = (n + 1)(n - 1), when n is odd and not a multiple of 3, we will have the product of two consecutive even integers, one of which is a multiple of 3, and thus n^2 - 1 is divisible by 24. For instance, when n is 5, we have 4 x 6, which is divisible by 24.

Statement One Alone:

n is a prime number.

When n is 5, we see that 24/24 = integer; however, when n = 2, 3/24 does not equal an integer. Statement one is not sufficient to answer the question.

Statement Two Alone:

n is greater than 191.

If n = 192, then n^2 - 1 will be odd and will not be divisible by 24. If n = 199, then n^2 - 1 = (199 + 1)(199 - 1) = 200 x 198 is divisible by 24, since 200 is divisible by 8 and 198 is divisible by 3. Statement two is not sufficient to answer the question.

Statements One and Two together:

From both statements, we see that n is a prime that is greater than 191, and thus it satisfies the case that n is odd and not a multiple of 3. So, n^2 - 1 will be divisible by 24.

Answer: C

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