Numbers Divisibility

A positive integer is said to be a prime number if it is not divisible by any positive integer other than itself and 1. Let p be a prime number greater than 5. Then (p2 - 1) is

  1. always divisible by 6, and may or may not be divisible by 12
  2. always divisible by 24
  3. never divisible by 6
  4. always divisible by 12, and may or may not be divisible by 24

Solution

If p is a prime number greater than 3, then p2-1 is always divisible by 24.

p2 - 1 = (p - 1) x (p + 1)

As p is a prime number, it must be odd. So, p - 1 and p + 1 must be even. Now, these 2 even numbers are consecutive. One of them must be a multiple of 4.

p - 1, p and p + 1 form three consecutive numbers. In any three consecutive numbers, one will be a multiple of 3. As p is not a multiple of 3 (p is prime), hence either p - 1 or p + 1 is a multiple. Therefore p2 - 1 has factors: 2, 4, & 3. 

Hence, p2 - 1 = 24n

When p = 7, p2 - 1 = 48

When p = 11, p2 - 1 = 120

The correct option is B.