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(n - 1) (n + 1) is divided by 24

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(n - 1) (n + 1) is divided by 24

by sanju09 » Sat Mar 19, 2011 1:34 am
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.




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by Anurag@Gurome » Sat Mar 19, 2011 2:05 am
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.
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