Question

use euclids division Lemma to show that the square of any positive integer is either of the form 3M or 3 m plus one for some integer m

Answer 1

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Answer 2

Let a be any positive integer

Let b = 3

Then a = 3q + r

a = 3q , a = 3q + 1 and a = 3q + 2

If a = 3q for some integer q

Then

a² => 9q² = 3(3q²)

a² = 3m (m = 3q² for some integer)

If a = 3q + 1 for some integer q

Then

a² => (3q + 1)² = 9q² + 6q + 1

= 3(3q² + 2q) + 1

= 3m + 1 (m = 3q² + 2q for some integer )

If a = 3q + 2 for some integer q

Then

a² => (3q + 2)² = 9q² + 12q + 3 + 1

= 3(3q² + 4q + 1) + 1

= 3m + 1 (m = 3q² - 4q + 1 for some integer)

Hence the square of any positive integer is of form 3m or 3m + 1 for some integer m