Data Sufficiency

Akansha wrote:If n is a positive integer and r is remainder when (n-1)(n+1) is divided by
24, what is the value of r?
a. n is divisible by 2
b. n is not divisible by 3


OA is C

Statement 1) should read as 'n is not divisible by 2'.
Solution:
Let us first consider statement 1) alone.
If we take n = 5 and n = 9, we will get two different values for r.
Or 1) alone is not sufficient to answer the question.
We next consider 2) alone.
If we take n = 2 and n = 4, we get two different values for r.
Or 2) alone is not sufficient to answer the question.
Next, combine both the statements together and check.
On combining, we have that n is of the form 6k+1 or 6k+5, k being 0,1,2,3....
If n = 6k+1, (n+1)(n-1) = (6k+2)(6k)= 12k(3k+1). If k is even, 12k is divisible by 24 and hence n = 12k(3k+1) is divisible by 24.
If k is odd, (3k+1) is even and hence n = 12k(3k+1) is divisible by 24.
If n = 6k+5, (n+1)(n-1) = (6k+6)(6k+4) = 12(k+1)(3k+2).
If k is even, 12(3k+2) is divisible by 24 and hence n = 12(k+1)(3k+2) is divisible by 24.
If k is odd, 12(k+1) is divisible by 24 and hence n = 12(k+1)(3k+2) is divisible by 24.
In either case r = 0.
So, both statements together are sufficient to answer the question.
The correct answer is (C).