How many divisors of 21600 are perfect squares ?

Answer

To find how many divisors of 21600 are perfect squares, we will follow these steps:

### Step 1: Prime Factorization of 21600
First, we need to factor 21600 into its prime factors.

1. Start with 21600 and divide by 10:
\[ 21600 = 216 \times 100 = 216 \times 10^2 \]
2. Factor 100:
\[ 100 = 10 \times 10 = 2^2 \times 5^2 \]
3. Now, factor 216:
\[ 216 = 2^3 \times 3^3 \]
(since \( 216 = 2 \times 108 = 2 \times 2 \times 54 = 2 \times 2 \times 2 \times 27 = 2^3 \times 3^3 \))

4. 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: Identify the Exponents
From the prime factorization, we have:
\[ 21600 = 2^5 \times 3^3 \times 5^2 \]

### Step 3: Determine Perfect Square Divisors
A divisor is a perfect square if all the exponents in its prime factorization are even.

1. For \(2^5\), the possible even exponents are 0, 2, and 4 (3 options).
2. For \(3^3\), the possible even exponents are 0 and 2 (2 options).
3. For \(5^2\), the possible even exponents are 0 and 2 (2 options).

### Step 4: Calculate the Total Number of Perfect Square Divisors
To find the total number of perfect square divisors, multiply the number of choices for each prime factor:
\[ \text{Total Perfect Square Divisors} = (3 \text{ options for } 2) \times (2 \text{ options for } 3) \times (2 \text{ options for } 5) \]
\[ = 3 \times 2 \times 2 = 12 \]

### Final Answer
Thus, the number of divisors of 21600 that are perfect squares is **12**.

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