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
OA C
(1) n is a prime number.
(2) n is greater than 191
OA C
Question rephrased: Is (n+1)(n-1) divisible by 24?guerrero 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
OA C
Statement 1: n is a prime number
If n=2, then (n+1)(n-1) = 3*1, which is NOT divisible by 24.
If n=23, then (n+1)(n-1) = 24*22, which IS divisible by 24.
INSUFFICIENT.
Statement 2: n is greater than 191
If n = 239, then (n+1)(n-1) = 240*238, which IS divisible by 24.
If n = 200, then (n+1)(n-1) = 201*199, which is ODD and thus NOT divisible by 24.
INSUFFICIENT.
Statements combined:
Rule 1: Of every 3 consecutive integers n-1, n, and n+1, exactly ONE will be a multiple of 3.
Since n is a prime number greater than 191, n must be an ODD integer that is NOT a multiple of 3.
Implication:
Either n-1 or n+1 MUST be a multiple of 3.
Rule 2: Of every 2 consecutive even integers, exactly one will be a multiple of 4, while the other will be an even integer that is not a multiple of 4.
Since n is odd, n-1 and n+1 are both EVEN.
Thus, either n-1 or n+1 is a multiple of 4, while the other is even but not a multiple of 4.
Implication:
(n-1)(n+1) must be a multiple of 4*2 = 8.
Result:
Since (n-1)(n+1) is a multiple of 3 and 8, it must be divisible by 24.
SUFFICIENT.
The correct answer is C.
