Find the number of divisors of 10800.

Options

A. 57
B. 60
C. 72
D. none of these

Answer

To find the number of divisors of 10800, we will follow these steps:

### Step 1: Prime Factorization of 10800
First, we need to perform the prime factorization of 10800.

1. Divide by 2:
- \( 10800 \div 2 = 5400 \)
- \( 5400 \div 2 = 2700 \)
- \( 2700 \div 2 = 1350 \)
- \( 1350 \div 2 = 675 \) (No more division by 2 possible)

2. Divide by 3:
- \( 675 \div 3 = 225 \)
- \( 225 \div 3 = 75 \)
- \( 75 \div 3 = 25 \) (No more division by 3 possible)

3. Divide by 5:
- \( 25 \div 5 = 5 \)
- \( 5 \div 5 = 1 \) (Finished)

Now we can express 10800 as a product of its prime factors:
\[ 10800 = 2^4 \times 3^3 \times 5^2 \]

### Step 2: Use the Formula for Finding the Number of Divisors
The formula for finding the number of divisors (d) of a number based on its prime factorization \( p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \) is:
\[ d = (e_1 + 1)(e_2 + 1)(e_3 + 1) \]

In our case:
- \( e_1 = 4 \) (for \( 2^4 \))
- \( e_2 = 3 \) (for \( 3^3 \))
- \( e_3 = 2 \) (for \( 5^2 \))

### Step 3: Calculate the Number of Divisors
Now substituting the exponents into the formula:
\[ d = (4 + 1)(3 + 1)(2 + 1) \]
\[ d = 5 \times 4 \times 3 \]

Calculating this step by step:
1. \( 5 \times 4 = 20 \)
2. \( 20 \times 3 = 60 \)

Thus, the number of divisors of 10800 is **60**.

### Final Answer
The number of divisors of 10800 is **60**.

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