Question

1) prove that the square of any positive integer is of the form 5q,5q + 1,5q+4for some integer q.

Answer 1

let a be any positive integer and b = 5

by EDL we get,

➡ a = bq + r where 0 ≤ r < b

so the possible values of r = 0, 1, 2, 3 and 4

when r = 0, a = 5q + 0

➡ a² = (5q)²

➡ a² = 25q²

➡ a² = 5(5q²)

when r = 1, a = 5q + 1

➡ a² = (5q + 1)²

➡ a² = 25q² + 10q + 1

➡ a² = 5(5q² + 2q) + 1

➡ a² = 5q + 1

when r = 2, a = 5q + 2

➡ a² = (5q + 2)²

➡ a² = 25q² + 20q + 4

➡ a² = 5(5q² + 4q) + 4

➡ a² = 5q + 4

when r = 3, a = 5q + 2l3

➡ a² = (5q + 3)²

➡ a² = 25q² + 30q + 5 + 4

➡ a² = 5(5q² + 6q + 1) + 4

➡ a² = 5q + 4

when r = 4, a = 5q + 4

➡ a² = (5q + 4)²

➡ a² = 25q² + 40q + 15 + 1

➡ a² = 5(5q² + 8q + 3) + 1

➡ a² = 5q + 1

hence, it's proved that the square of any positive integer is of the form 5q, 5q + 1, 5q+4 for some integer q.