Use divison algorithm to show that the square of any positive integer is of the form 3p or 3p+1?

Question

Answer

Using Division Algorithm to Prove the Statement

Divison Algorithm:

The Division Algorithm states that for any two integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < />

Proof:

Case 1: Square of an integer is of the form 3p

Let's consider a positive integer n and its square n^2.
When we divide n^2 by 3, using the Division Algorithm, we get n^2 = 3p for some integer p.
Therefore, the square of any positive integer can be represented as 3p, where p is an integer.

Case 2: Square of an integer is of the form 3p + 1

If n is not divisible by 3, then n can be represented as 3q + 1 or 3q + 2 for some integer q.
When we square n, we get n^2 = (3q + 1)^2 = 3(3q^2 + 2q) + 1 = 3p + 1, where p = 3q^2 + 2q is an integer.
Therefore, the square of any positive integer can also be represented as 3p + 1.