Prove that the square of any positive integer is of the form 3m or, 3m

1.	Prove that the square of any positive integer is of the form 3m or, 3m + 1 but not of the form 3m + 2.
Answer» we know, That any positive integer N is of the form 3q, 3q + 1 or, 3q + 2. When N = 3q, then N² = 9q² = 3(3q)² = 3m where m = 3q² And, as q is an integer, m = 3q² is also an integer. When N = 3q + 1, then N² = (3q + 1)² = 9q² + 6q + 1 ⇒ 3q(3q + 2) + 1 = 3m + 1 where m = q(3q + 2) And, as q is an integer, m = q(3q + 2) is also an integer. When N = 3q + 2, then N² = (3q + 2)² = 9q²+ 12q + 4 ⇒ 3(3q² + 4q + 1) + 1 3m + 1 where, m = 3q²+ 4q + 1 And, as q is an integer, m = 3q²+ 4q + 1 is also an integer. Therefore, N² is of the form 3m, 3m + 1 but not of the form 3m + 2

Prove that the square of any positive integer is of the form 3m or, 3m + 1 but not of the form 3m + 2.

Answer»

we know,

That any positive integer N is of the form 3q, 3q + 1 or, 3q + 2.

When N = 3q, then

N² = 9q² = 3(3q)² = 3m where m = 3q²

And,

as q is an integer,

m = 3q² is also an integer.

When N = 3q + 1,

then N² = (3q + 1)² = 9q² + 6q + 1

⇒ 3q(3q + 2) + 1 = 3m + 1

where m = q(3q + 2)

And,

as q is an integer,

m = q(3q + 2) is also an integer.

When N = 3q + 2,

then N² = (3q + 2)² = 9q²+ 12q + 4

⇒ 3(3q² + 4q + 1) + 1 3m + 1

where,

m = 3q²+ 4q + 1

And,

as q is an integer,

m = 3q²+ 4q + 1 is also an integer.

Therefore,

N² is of the form 3m, 3m + 1

but not of the form 3m + 2