The unit digit of the product of n consecutive numbers of the form 2k+1, where `k in I^+` and n must be at least 5.

Options

A. zero
B. any odd number
C. 5
D. BOTH (b) AND (c)

Answer

To determine the unit digit of the product of n consecutive numbers of the form \(2k + 1\) (where \(k\) is a positive integer and \(n \geq 5\)), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Form of Numbers**:
The expression \(2k + 1\) generates odd numbers for any positive integer \(k\). For example, if \(k = 1\), we get \(3\); if \(k = 2\), we get \(5\); and so on.

2. **Identifying n Consecutive Odd Numbers**:
If we take \(n\) consecutive odd numbers starting from \(2k + 1\), we can express them as:
- First number: \(2k + 1\)
- Second number: \(2k + 3\)
- Third number: \(2k + 5\)
- Fourth number: \(2k + 7\)
- Fifth number: \(2k + 9\)
- And so on...

For \(n = 5\), the numbers would be \(2k + 1, 2k + 3, 2k + 5, 2k + 7, 2k + 9\).

3. **Finding the Unit Digits**:
The unit digits of these numbers will depend on the value of \(k\):
- If \(k = 1\), the numbers are \(3, 5, 7, 9, 11\) (unit digits: 3, 5, 7, 9, 1).
- If \(k = 2\), the numbers are \(5, 7, 9, 11, 13\) (unit digits: 5, 7, 9, 1, 3).
- If \(k = 3\), the numbers are \(7, 9, 11, 13, 15\) (unit digits: 7, 9, 1, 3, 5).
- If \(k = 4\), the numbers are \(9, 11, 13, 15, 17\) (unit digits: 9, 1, 3, 5, 7).

4. **Calculating the Product**:
Regardless of the starting point, we will always include the number \(5\) in the product when \(n \geq 5\). The presence of \(5\) ensures that the unit digit of the product will be determined by the unit digit of \(5\) multiplied by the unit digits of the other odd numbers.

5. **Conclusion**:
Since the product of any odd number with \(5\) will always have a unit digit of \(5\), we conclude that the unit digit of the product of \(n\) consecutive odd numbers (where \(n \geq 5\)) is always \(5\).

### Final Answer:
The unit digit of the product of \(n\) consecutive numbers of the form \(2k + 1\) (where \(k\) is a positive integer and \(n \geq 5\)) is **5**.