QuestionShow that any positive odd integer is of the form 6q+1, or 6q+3, or 6q+5, where q is some integer.HardHide solution>

1. Break Down the Problem

We need to prove that any positive odd integer can be expressed in the forms $6q + 1$, $6q + 3$, or $6q + 5$, where $q$ is some integer.

2. Relevant Concepts

• Odd Integers: An integer is odd if it can be expressed in the form $2k + 1$ for some integer $k$.
• Modulo Operation: Any integer can be expressed in terms of its remainder when divided by another integer (in this case, 6).

3. Analysis and Detail

1. Odd Integer Representation: Any positive odd integer can be written in the form: $n = 2k + 1$ for some integer $k$.

2. Expressing in Terms of Modulo 6: Next, we can consider the possible values of $n$ when divided by 6:

   • The complete residue system modulo 6 is: $0, 1, 2, 3, 4, 5$.
   • The odd integers among these residues are $1, 3, 5$.

3. Checking Residue Classes:

   • Case $6q + 1$: Any integer of the form $6q + 1$ is odd.
   • Case $6q + 3$: Similarly, any integer of the form $6q + 3$ is odd.
   • Case $6q + 5$: Finally, any integer of the form $6q + 5$ is also odd.

4. Verify and Summarize

Since we have shown that all possible odd integers modulo 6 can be represented by one of the forms $6q + 1$, $6q + 3$, or $6q + 5$, it is confirmed that any positive odd integer can indeed be expressed in one of these forms.

Final Answer

Thus, any positive odd integer is of the form $6q + 1$, $6q + 3$, or $6q + 5$, where $q$ is some integer.