Use proof by contradiction to prove that if for an integer a, a² is divisible by 3, then a is divisible by 3.

By seeking contradiction,

Let a² is divisible by 3 and a is not divisible by 3.

So, a can be of the form :

⇒ a = 3k + 1 and a = 3k + 2

⇒ a² = (3k + 1)² or a² = (3k + 2)²

⇒ a² = 9k² + 6k + 1 or a² = 9k² + 12k + 4

⇒ a² = 3(3k² + 2k) + 1 or a² = 3(3k² + 4k) + 4

Let 3k² + 2k = c and 3k² + 4k = d.

a² = 3c + 1 or a² = 3d + 4

We know for any integer c and d 3c + 1 and 3d + 4 is not divisible by 3.

So, a² will not be divisible by 3.

So, our assumption is wrong

∴ a will be divisible by 3.

Hence, proved that if for an integer a, a² is divisible by 3, then a is divisible by 3.