If n is a positive integer, is n^2-1 divisible by 24?
A: n is a prime number
B: n is greater than 191
A: n is a prime number
B: n is greater than 191
Case A: n is a prime number
If n=2, (2)^2-1= 4-1= 3 => Not Divisible by 24
If n=5, (5)^2-1= 24 => Divisible by 24.
Thus we do not get a unique value. Therefore, NOT SUFFICIENT
Case B: n>191
Not sufficient as we get different answers for n=24^2 and n=17^2
Combined:
n is prime and greater than 191.
Thus, n is odd and not a multiple of 3.
As n^2-1 = (n-1)(n+1), either of (n-1),(n+1) is a multiple of 3.
Also, as n is odd, (n-1) and (n+1) are even.
So, one of them must be a multiple of 4. Thus, (n-1)(n+1)=2*4=8.
Thus, for a number to be divisible by 24, it should be divisible by 8 & 3.
Thus answer is (C).
