Question

Use Euclid’s Division lemma to show that the square of any positive integer is either of the form 3m, 3m+1 for some integer m.

Answer 1

Let a be any positive integer. Let q be the quotient and r be remainder. Then a = bq + r where q and r are also positive integers and 0 ≤ r < b

Taking b = 3, we get

a = 3q + r; where 0 ≤ r < 3

When, r = 0 = ⇒ a = 3q

When, r = 1 = ⇒ a = 3q + 1

When, r = 2 = ⇒ a = 3q + 2

Now, we have to show that the squares of positive integers 3q, 3q + 1 and 3q + 2 can be expressed as 3m or 3m + 1 for some integer m.

⇒ Squares of 3q = (3q)²

= 9q² = 3(3q)² = 3 m where m is some integer.

Square of 3q + 1 = (3q + 1)²

= 9q² + 6q + 1 = 3(3q² + 2 q) + 1

= 3m +1, where m is some integer

Square of 3q + 2 = (3q + 2)²

= (3q + 2)²

= 9q² + 12q + 4

= 9q² + 12q + 3 + 1

= 3(3q² + 4q + 1)+ 1

= 3m + 1 for some integer m.

∴ The square of any positive integer is either of the form 3m or 3m + 1 for some integer m.