In this problem, we study the form of the square of a positive integer. We are required to prove that the square of any positive integer can always be written in the form 4q or 4q + 1 for some integer q.
Question
Prove that the square of any positive integer is of the form 4q or 4q + 1 for some integer q.
Solution
Let n be any positive integer.
Every positive integer can be written in one of the following two forms:
4q or 4q + 1 or 4q + 2 or 4q + 3, where q is an integer.
Now, we consider each case.
Case 1: n = 4q
Then,
n² = (4q)²
= 16q²
= 4(4q²)
So, n² is of the form 4q.
Case 2: n = 4q + 1
Then,
n² = (4q + 1)²
= 16q² + 8q + 1
= 4(4q² + 2q) + 1
So, n² is of the form 4q + 1.
Case 3: n = 4q + 2
Then,
n² = (4q + 2)²
= 16q² + 16q + 4
= 4(4q² + 4q + 1)
So, n² is of the form 4q.
Case 4: n = 4q + 3
Then,
n² = (4q + 3)²
= 16q² + 24q + 9
= 4(4q² + 6q + 2) + 1
So, n² is of the form 4q + 1.
From all the above cases, we see that the square of a positive integer is always of the form 4q or 4q + 1.
Conclusion
Therefore, the square of any positive integer is of the form 4q or 4q + 1 for some integer q.