sanju09 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: Let's take two values for n as follows,

• 1. n = 1 --> (n - 1)(n + 1) = 0 --> Remainder = 0
  2. n = 3 --> (n - 1)(n + 1) = 8 --> Remainder = 8

Not sufficient

Statement 2: Let's take two values for n as follows,

• 1. n = 1 --> (n - 1)(n + 1) = 0 --> Remainder = 0
  2. n = 2 --> (n - 1)(n + 1) = 3 --> Remainder = 3

Not sufficient

1 & 2 Together: Now as n is odd, both (n - 1) and (n + 1) must be even. And as (n - 1) and (n + 1) are consecutive even integers, any one of them must be divisible by 4. Hence, the product (n - 1)(n + 1) must be divisible by 2*4 = 8.

Again (n - 1), n, and (n + 1) are three consecutive integers.
Hence, any one of them must be divisible by 3. As n is not divisible by 3, either (n - 1) or (n + 1) must be.

Hence, the product (n - 1)(n + 1) will always be divisible by 3 and 8, i.e. 24

Sufficient

the correct answer is C.