Find the number of divisors of 21600.

Answer

To find the number of divisors of 21600, we will follow these steps:

### Step 1: Prime Factorization of 21600
We start by performing the prime factorization of 21600. We can break it down as follows:
- \( 21600 = 216 \times 100 \)
- \( 216 = 6^3 = (2 \times 3)^3 = 2^3 \times 3^3 \)
- \( 100 = 10^2 = (2 \times 5)^2 = 2^2 \times 5^2 \)

Now, combining these factors:
\[ 21600 = 2^3 \times 3^3 \times 2^2 \times 5^2 \]

### Step 2: Combine the Prime Factors
Next, we combine the powers of the prime factors:
\[ 2^3 \times 2^2 = 2^{3+2} = 2^5 \]
Thus, we can rewrite 21600 as:
\[ 21600 = 2^5 \times 3^3 \times 5^2 \]

### Step 3: Use the Divisor Formula
The formula to find the number of divisors \( d(n) \) of a number \( n \) with the prime factorization \( p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \) is given by:
\[ d(n) = (e_1 + 1)(e_2 + 1)(e_3 + 1) \]

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

### Step 4: Calculate the Number of Divisors
Now we can apply the formula:
\[ d(21600) = (5 + 1)(3 + 1)(2 + 1) \]
Calculating each term:
\[ d(21600) = 6 \times 4 \times 3 \]

### Step 5: Final Calculation
Now we perform the multiplication:
\[ 6 \times 4 = 24 \]
\[ 24 \times 3 = 72 \]

Thus, the number of divisors of 21600 is **72**.

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