A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, 3m or 3m + 2? Justify your answer.

Introduction

In this problem, we examine the square of a positive integer of the form 3q + 1. We will check whether its square can be written in any form other than 3m + 1, 3m, or 3m + 2 for some integer m.

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Question

A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, 3m, or 3m + 2 for some integer m? Justify your answer.

Solution

Let the given positive integer be of the form 3q + 1, where q is a natural number.

Now, consider the square of the given integer.

(3q + 1)² = 9q² + 6q + 1

This expression can be written as:

9q² + 6q + 1 = 3(3q² + 2q) + 1

Since q is a natural number, the expression (3q² + 2q) is an integer. Let 3q² + 2q = m, where m is an integer.

Therefore, the square of the given number can be written as:

3m + 1

Thus, the square of a positive integer of the form 3q + 1 is always of the form 3m + 1 and cannot be written in the form 3m or 3m + 2.

Conclusion

Hence, the square of a positive integer of the form 3q + 1 can only be written in the form 3m + 1 for some integer m, and not in any other form.

Hence proved.

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