Show that the square of an odd positive integer can be of the form 6q + 1 or 6q + 3 for some integer q

Introduction

In this problem, we study the form of the square of an odd positive integer. We will show that the square of an odd positive integer can be written in the form 6q + 1 or 6q + 3 for some integer q.

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Question

Show that the square of an odd positive integer can be of the form 6q + 1 or 6q + 3 for some integer q.

Solution

Let n be any odd positive integer. Then n can be written in the form 2k + 1, where k is an integer.

Now, consider the square of n.

n² = (2k + 1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1.

Every integer can be written in one of the following three forms: 3m, 3m + 1, or 3m + 2, where m is an integer.

So, the value of 2k² + 2k can be written as 3m or 3m + 1 or 3m + 2.

Therefore, n² can be written as:

If 2k² + 2k = 3m, then n² = 6m + 1.

If 2k² + 2k = 3m + 1, then n² = 6m + 3.

If 2k² + 2k = 3m + 2, then n² = 6m + 5.

But an odd integer squared cannot be of the form 6q + 5. Hence, the square of an odd positive integer can only be of the form 6q + 1 or 6q + 3.

Conclusion

Therefore, the square of an odd positive integer can be written in the form 6q + 1 or 6q + 3 for some integer q.

Hence proved.