Question

Show that the square of any positive integer of the form 3m or 3m + 1 for some integer m.

Answer 1

Let a be any positive integer. Then it is of the form 3q or 3q + 1 or 3q + 2 

Case – 1 When a = 3q 

a² = (3q)² = 9q²

= (3q) (3q) 

= 3m where m = 3q 

Case – 2 When a = 3q + 1 

a² = (3q + 1)² = 9q + 6q + 1 

= 3q (3q + 2) + 1 

= 3 m + 1

where m = q (3q + 2) 

Case – 3 When a = 3q + 2 

a² = (3q + 2)² = 9q² + 12q + 4 

= 9q² + 12q + 3 + 1 

= 3(3q² + 4q + 1) + 1 

= 3m + 1 

where m = 3q² + 4q + 1

Hence a is of the form 3m or 3m + 1