Data Sufficiency

If n is a positive integer and r is the remainder when (n-1)(n+1) is divided by 24, what is the value of r?

1). n is not divisible by 2

2). n is not divisible by 3

OE C

Well I can get the answer by taking numbers but can any1 explain it practically.

akash singhal wrote:If n is a positive integer and r is the remainder when (n-1)(n+1) is divided by 24, what is the value of r?

1). n is not divisible by 2

2). n is not divisible by 3

Statement 1: n is not divisible by 2
Case 1: n=1
Here, (n-1)(n+1)/24 = (0*2)/24 = 0/24 = 0 R0.
Case 2: n=3
Here, (n-1)(n+1)/34 = (2*4)/24 = 3/24 = 0 R3.
Since different remainders are possible, INSUFFICIENT.

Statement 2: n is not divisible by 3
Case 1 also satisfies Statement 2.
In Case 1, R=0.
Case 3: n=4
Here, (n-1)(n+1)/34 = (3*5)/24 = 15/24 = 0 R15.
Since different remainders are possible, INSUFFICIENT.

Statements combined:
n is equal to an ODD integer NOT DIVISIBLE BY 3:
1, 5, 7, 11, 13...

As shown above, n=1 results in R=0.
Case 4: n=5
Here, (n-1)(n+1)/34 = (4*6)/24 = 24/24 = 1 R0.
Case 5: n=7
Here, (n-1)(n+1)/34 = (6*8)/24 =48/24 = 2 R0.

In every case, R=0.
SUFFICIENT.

The correct answer is C.

Conceptual explanation for the combined statements:

n-1, n, n+1 are 3 consecutive integers.

Of every 3 consecutive integers, exactly 1 will be a multiple of 3.
Since n is NOT divisible by 3, either n-1 or n+1 must be a multiple of 3.

Since n = odd, n-1 and n+1 are two consecutive EVEN integers.
Of every two consecutive even integers, exactly one will be a multiple of 4.
Thus, (n-1)(n+1) = (multiple of 4)(even non-multiple of 4) = multiple of 8.

Since (n-1)(n+1) is divisible by both 3 and 8, (n-1)(n+1) is a multiple of 24.
Thus, dividing (n-1)(n+1) by 24 will yield a remainder of 0.