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

Options

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

Explanation

To solve the problem, we need to determine the form of every odd integer based on the given integer `q`. We will analyze each option provided in the question.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that `q` is an integer. We need to find a mathematical expression that represents every odd integer.

2. **Identifying Odd Integers**:
Odd integers can be expressed in the form of `2n + 1`, where `n` is any integer. This is because when you multiply an integer by 2, you get an even number, and adding 1 to it gives you an odd number.

3. **Analyzing the Options**:
We have four options to analyze:
- Option 1: `q`
- Option 2: `q + 1`
- Option 3: `2q`
- Option 4: `2q + 1`

4. **Evaluating Each Option**:
- **Option 1: `q`**
- If `q` is any integer, it can be either odd or even. Thus, this option does not represent only odd integers.

- **Option 2: `q + 1`**
- If `q` is an integer, `q + 1` will also be either odd or even depending on whether `q` is odd or even. Therefore, this option does not represent only odd integers.

- **Option 3: `2q`**
- The expression `2q` will always yield an even integer for any integer `q`. Thus, this option cannot represent odd integers.

- **Option 4: `2q + 1`**
- This expression will always yield an odd integer because it is in the form of `2n + 1`, where `n` is equal to `q`. Therefore, this option correctly represents every odd integer.

5. **Conclusion**:
The correct form that represents every odd integer is **Option 4: `2q + 1`**.

### Final Answer:
Every odd integer can be expressed in the form of `2q + 1`, where `q` is an integer.