For some integer `q`, every odd integer is of the form

Options

A. `q`
B. `q+1`
C. `2q`
D. `2q+1`

Explanation

To find a general form for every odd integer in terms of some integer \( q \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Odd and Even Numbers**:
- An even number can be expressed in the form \( 2n \), where \( n \) is an integer. For example, \( 0, 2, 4, 6, \ldots \) can be represented as \( 2 \times 0, 2 \times 1, 2 \times 2, 2 \times 3, \ldots \).
- An odd number can be expressed as an even number plus one. Therefore, if \( 2n \) is an even number, then \( 2n + 1 \) is an odd number.

2. **Defining \( q \)**:
- Let \( q \) be any integer. This means \( q \) can take values like \( 0, 1, -1, 2, -2, \ldots \).

3. **Expressing Even Numbers**:
- For any integer \( q \), the expression \( 2q \) will yield an even number. For example:
- If \( q = 0 \), then \( 2q = 0 \) (even).
- If \( q = 1 \), then \( 2q = 2 \) (even).
- If \( q = 2 \), then \( 2q = 4 \) (even).

4. **Finding the General Form for Odd Integers**:
- Since we know that every odd integer can be represented as an even integer plus one, we can write:
\[ \text{Odd Integer} = 2q + 1 \]
- This means that for any integer \( q \), \( 2q + 1 \) will always yield an odd integer. For example:
- If \( q = 0 \), then \( 2(0) + 1 = 1 \) (odd).
- If \( q = 1 \), then \( 2(1) + 1 = 3 \) (odd).
- If \( q = 2 \), then \( 2(2) + 1 = 5 \) (odd).

5. **Conclusion**:
- Therefore, we can conclude that every odd integer can be expressed in the form:
\[ \text{Every odd integer} = 2q + 1 \quad \text{for some integer } q. \]