Understanding Odd Positive Integers
An odd positive integer can be expressed in the form of \( 2k + 1 \), where \( k \) is a non-negative integer. The square of this integer will help us determine the required forms.
Calculating the Square
Let's calculate the square of an odd integer:
- If \( n = 2k + 1 \), then:
\[
n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 4k(k + 1) + 1
\]
- Notice that \( k(k + 1) \) is always even since it is the product of two consecutive integers.
Expressing in Terms of 6
Now, let’s analyze \( n^2 \) modulo 6:
- Any integer can be expressed as \( 6q + r \), where \( r \) is the remainder when divided by 6. The possible values for \( r \) are 0, 1, 2, 3, 4, or 5.
Possible Remainders for Odd Squares
Let’s consider the odd integers modulo 6:
- Odd integers are: \( 1, 3, 5 \).
1. For \( n \equiv 1 \mod 6 \):
\[
n^2 \equiv 1^2 \equiv 1 \mod 6
\]
2. For \( n \equiv 3 \mod 6 \):
\[
n^2 \equiv 3^2 \equiv 9 \equiv 3 \mod 6
\]
3. For \( n \equiv 5 \mod 6 \):
\[
n^2 \equiv 5^2 \equiv 25 \equiv 1 \mod 6
\]
Conclusion
From our calculations, \( n^2 \) can be expressed as:
- \( n^2 \equiv 1 \mod 6 \) (which can be represented as \( 6q + 1 \))
- \( n^2 \equiv 3 \mod 6 \) (which can be represented as \( 6q + 3 \))
Thus, the square of any odd positive integer can indeed be in the form of \( 6q + 1 \) or \( 6q + 3 \) for some integer \( q \).