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 be written only in the forms 5q, 5q + 1, or 5q + 4 for some integer q.
Question
Prove that the square of any positive integer is of the form 5q, 5q + 1, or 5q + 4 for some integer q.
Solution
Let n be any positive integer.
Every positive integer can be written in one of the following five forms:
5q, 5q + 1, 5q + 2, 5q + 3, or 5q + 4, where q is an integer.
Now, we consider each case.
Case 1: n = 5q
Then,
n² = (5q)²
= 25q²
= 5(5q²)
So, n² is of the form 5q.
Case 2: n = 5q + 1
Then,
n² = (5q + 1)²
= 25q² + 10q + 1
= 5(5q² + 2q) + 1
So, n² is of the form 5q + 1.
Case 3: n = 5q + 2
Then,
n² = (5q + 2)²
= 25q² + 20q + 4
= 5(5q² + 4q) + 4
So, n² is of the form 5q + 4.
Case 4: n = 5q + 3
Then,
n² = (5q + 3)²
= 25q² + 30q + 9
= 5(5q² + 6q + 1) + 4
So, n² is of the form 5q + 4.
Case 5: n = 5q + 4
Then,
n² = (5q + 4)²
= 25q² + 40q + 16
= 5(5q² + 8q + 3) + 1
So, n² is of the form 5q + 1.
From all the above cases, we see that the square of a positive integer is always of the form 5q, 5q + 1, or 5q + 4.
Conclusion
Therefore, the square of any positive integer is of the form 5q, 5q + 1, or 5q + 4 for some integer q.
Hence proved.