How many natural factors does 26,460 have?

To find the number of natural factors of 26,460, we first need to perform its prime factorization.

### Step 1: Prime Factorization of 26,460

We will start dividing 26,460 by the smallest prime number and continue dividing until we can’t anymore.

1. **Divide by 2:**  
   \( 26,460 \div 2 = 13,230 \)
   
2. **Divide by 2:**  
   \( 13,230 \div 2 = 6,615 \)

3. **Divide by 3:**  
   \( 6,615 \div 3 = 2,205 \)

4. **Divide by 3:**  
   \( 2,205 \div 3 = 735 \)

5. **Divide by 5:**  
   \( 735 \div 5 = 147 \)

6. **Divide by 3:**  
   \( 147 \div 3 = 49 \)

7. **Divide by 7:**  
   \( 49 \div 7 = 7 \)

8. **Divide by 7:**  
   \( 7 \div 7 = 1 \)

So, the prime factorization of 26,460 is:

\[
26,460 = 2^2 \times 3^3 \times 5 \times 7^2
\]

### Step 2: Applying the Formula for Number of Factors

The formula to find the total number of factors is:

\[
\text{Number of factors} = (e_1 + 1)(e_2 + 1)(e_3 + 1) \cdots (e_n + 1)
\]

Where \( e_1, e_2, e_3, \dots \) are the exponents of the prime factors.

From the prime factorization \( 2^2 \times 3^3 \times 5^1 \times 7^2 \):

\[
\text{Number of factors} = (2+1)(3+1)(1+1)(2+1)
\]

\[
= 3 \times 4 \times 2 \times 3 = 72
\]

### Final Answer:
The number of natural factors of 26,460 is **72**.

Would you like more details or have any questions?

Here are five related questions to explore:
1. What are the prime factors of 12,345?
2. How do you find the greatest common divisor (GCD) of two numbers using prime factorization?
3. How can you find the least common multiple (LCM) of two numbers?
4. How many natural factors does 50,000 have?
5. What is the sum of the natural factors of a number?

**Tip:** The number of factors of a number depends directly on the exponents in its prime factorization!

Learn how to find the number of natural factors of 26,460 with a step-by-step guide using prime factorization. This problem explores number theory concepts and is suitable for students in Grades 8-10.

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