The largest number which always divides the sum of any pair of consecutive odd numbers is

Options

A. 2
B. 4
C. 6
D. 8

Answer

To solve the problem of finding the largest number that always divides the sum of any pair of consecutive odd numbers, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Odd Numbers**:
Odd numbers are numbers that are not divisible by 2. The sequence of consecutive odd numbers can be represented as:
- First odd number: \(2n + 1\)
- Second consecutive odd number: \(2n + 3\)

2. **Finding the Sum of Consecutive Odd Numbers**:
We need to find the sum of these two consecutive odd numbers:
\[ (2n + 1) + (2n + 3) \]

3. **Simplifying the Sum**:
Combine the terms:
\[ 2n + 1 + 2n + 3 = 4n + 4 \]

4. **Factoring the Sum**:
Factor out the common term from the expression:
\[ 4n + 4 = 4(n + 1) \]

5. **Identifying the Divisor**:
The expression \(4(n + 1)\) shows that the sum of any pair of consecutive odd numbers is always divisible by 4, regardless of the value of \(n\).

6. **Conclusion**:
Therefore, the largest number that always divides the sum of any pair of consecutive odd numbers is:
\[ \boxed{4} \]