How many divisors of 21600 are even numbers?

Answer

To find how many divisors of 21600 are even numbers, we can follow these steps:

### Step 1: Prime Factorization of 21600
First, we need to perform the prime factorization of 21600.

1. **Divide by 10**:
\[ 21600 = 216 \times 100 \]
\[ 100 = 10^2 = (2 \times 5)^2 = 2^2 \times 5^2 \]

2. **Factor 216**:
\[ 216 = 2^3 \times 3^3 \]

3. **Combine the factors**:
\[ 21600 = 2^3 \times 3^3 \times 2^2 \times 5^2 = 2^{3+2} \times 3^3 \times 5^2 = 2^5 \times 3^3 \times 5^2 \]

### Step 2: Calculate Total Number of Divisors
Using the formula for the number of divisors based on prime factorization, where if \( n = p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \), then the total number of divisors \( D(n) \) is given by:
\[ D(n) = (e_1 + 1)(e_2 + 1)(e_3 + 1) \]

For \( 21600 = 2^5 \times 3^3 \times 5^2 \):
- \( e_1 = 5 \) (for 2)
- \( e_2 = 3 \) (for 3)
- \( e_3 = 2 \) (for 5)

Calculating the total number of divisors:
\[ D(21600) = (5 + 1)(3 + 1)(2 + 1) = 6 \times 4 \times 3 = 72 \]

### Step 3: Calculate Number of Odd Divisors
Odd divisors are those that do not include the factor of 2. Therefore, we only consider the factors of \( 3^3 \) and \( 5^2 \).

For \( 3^3 \times 5^2 \):
- \( e_2 = 3 \) (for 3)
- \( e_3 = 2 \) (for 5)

Calculating the number of odd divisors:
\[ D_{\text{odd}} = (3 + 1)(2 + 1) = 4 \times 3 = 12 \]

### Step 4: Calculate Number of Even Divisors
To find the number of even divisors, we subtract the number of odd divisors from the total number of divisors:
\[ D_{\text{even}} = D(21600) - D_{\text{odd}} = 72 - 12 = 60 \]

### Final Answer
Thus, the number of even divisors of 21600 is **60**.

---