Show that the square of any positive integer cannot be of the form 6q + 2 or 6q + 5 for any integer q

Introduction

In this problem, we study the possible forms of the square of a positive integer. We will show that the square of any positive integer can never be written in the form 6q + 2 or 6q + 5 for any integer q.

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Question

Show that the square of any positive integer cannot be of the form 6q + 2 or 6q + 5 for any integer q.

Solution

Let n be any positive integer. Every positive integer can be written in one of the following forms:

6q, 6q + 1, 6q + 2, 6q + 3, 6q + 4, or 6q + 5, where q is an integer.

If n = 6q, then n² = (6q)² = 36q², which is divisible by 6.

If n = 6q + 1, then n² = (6q + 1)² = 36q² + 12q + 1, which is of the form 6q + 1.

If n = 6q + 2, then n² = (6q + 2)² = 36q² + 24q + 4, which is of the form 6q + 4.

If n = 6q + 3, then n² = (6q + 3)² = 36q² + 36q + 9, which is of the form 6q + 3.

If n = 6q + 4, then n² = (6q + 4)² = 36q² + 48q + 16, which is of the form 6q + 4.

If n = 6q + 5, then n² = (6q + 5)² = 36q² + 60q + 25, which is of the form 6q + 1.

From the above cases, we see that the square of a positive integer can be of the form 6q, 6q + 1, 6q + 3, or 6q + 4 only.

Hence, the square of any positive integer cannot be of the form 6q + 2 or 6q + 5.

Conclusion

Therefore, the square of any positive integer cannot be written in the form 6q + 2 or 6q + 5 for any integer q.

Hence proved.

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