Question

Use EDL to show that the square of any positive integer is either of the form of 3n or 3n+1 for some integer..​

Answer 1

Step-by-step explanation:

what is the full form of EDL

Answer 2

let us start by taking 'a' any positive integer and b = 3

by Euclid's division lemma,

a = bq + r where 0 ≤ r < b

therefore r = 0, 1 or 2

when r = 0, a = 3q ---------(i)

when r = 1, a = 3q + 1 ---------(ii)

when r = 2, a = 3q + 2 ---------(iii)

squaring both sides of each equation.

(i) a² = (3q)²

➡ a² = 9q²

➡ a² = 3(3q²)

➡ a² = 3m --------(iv)

(ii) a² = (3q + 1)²

➡ a² = 9q² + 6q + 1

➡ a² = 3(3q² + 2q) + 1

➡ a² = 3m + 1 ---------(v)

(iii) a² = (3q + 2)²

➡ a² = 9q² + 12q + 4

➡ a² = 9q² + 12q + 3 + 1

➡ a² = 3(3q² + 4q + 1) + 1

➡ a² = 3m + 1 ---------(vi)

by equation (iv), (v) and (vi) it's proved that the square of any positive number is either in the form of 3m or 3m + 1.