Which one of the following is a factor of the sum of first twenty-five natural numbers ?

Options

A. 26
B. 24
C. 13
D. 12

Answer

To find which one of the following is a factor of the sum of the first twenty-five natural numbers, we can follow these steps:

### Step-by-step Solution:

1. **Understand the Problem**: We need to find the sum of the first 25 natural numbers and then determine which of the given options is a factor of that sum.

2. **Use the Formula for the Sum of Natural Numbers**: The formula for the sum of the first \( n \) natural numbers is given by:
\[ S_n = \frac{n(n + 1)}{2} \]
Here, \( n = 25 \).

3. **Calculate the Sum**:
\[ S_{25} = \frac{25(25 + 1)}{2} = \frac{25 \times 26}{2} \]
Simplifying this:
\[ S_{25} = \frac{650}{2} = 325 \]

4. **Identify the Factors of 325**: Next, we need to find the factors of 325. To do this, we can start by finding its prime factorization:
- 325 can be divided by 5 (since it ends in 5):
\[ 325 \div 5 = 65 \]
- Now, factor 65:
\[ 65 \div 5 = 13 \]
- Therefore, the prime factorization of 325 is:
\[ 325 = 5^2 \times 13 \]

5. **List the Factors**: The factors of 325 can be determined from its prime factorization:
- The factors are: \( 1, 5, 13, 25, 65, 325 \).

6. **Check the Given Options**: Now, we need to check which of the given options is a factor of 325. If one of the options is 13, then it is a factor.

7. **Conclusion**: Since 13 is indeed one of the factors of 325, the answer to the question is:
\[ \text{The factor of the sum of the first twenty-five natural numbers is } 13. \]