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 in the form 3m or 3m + 1, but never in the form 3m + 2.
Question
Prove that the square of any positive integer is of the form 3m or 3m + 1 but not of the form 3m + 2.
Solution
Let n be any positive integer.
Every positive integer can be written in one of the following three forms:
3m, 3m + 1, or 3m + 2, where m is an integer.
Case 1: n = 3m
Then,
n² = (3m)² = 9m² = 3(3m²)
So, n² is of the form 3m.
Case 2: n = 3m + 1
Then,
n² = (3m + 1)²
= 9m² + 6m + 1
= 3(3m² + 2m) + 1
So, n² is of the form 3m + 1.
Case 3: n = 3m + 2
Then,
n² = (3m + 2)²
= 9m² + 12m + 4
= 3(3m² + 4m + 1) + 1
So, n² is also of the form 3m + 1.
From all the above cases, we see that the square of a positive integer is always of the form 3m or 3m + 1, and never of the form 3m + 2.
Conclusion
Therefore, the square of any positive integer is of the form 3m or 3m + 1, but not of the form 3m + 2.
Hence proved.