If p ≥ 5 is a prime number, show that p² + 2 is divisible by 3.

Given,

p ≥ 5

We know that,

Every prime number greater than 3 is of the form 6k + 1 or 6k + 5.

Let p be of form (6k + 1)

p² + 2 = (6k + 1)² + 2

= 36k² + 1 + 12k + 2

= 36k² + 3 + 12k

= 3(12k² + 1 + 4k); which is clearly divisible by 3.

Let p be of the form (6k + 5)

p² + 2 = (6k + 5)² + 2

= 36k² + 25 + 60k + 2

= 36k² + 60k + 27

= 3(12k² + 20k + 9); which is clearly divisible by 3.

Hence, proved that for p being prime number and p ≥ 5, p² + 2 is divisible by 3.