How many divisors of 21600 are odd numbers?

Answer

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

### Step 1: Prime Factorization of 21600
First, we need to find the prime factorization of 21600. We can break it down as follows:

\[ 21600 = 216 \times 100 \]

Next, we can factor 216 and 100:

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

- **Factor 100:**
\[ 100 = 10^2 = (2 \times 5)^2 = 2^2 \times 5^2 \]

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

### Step 2: Identify Odd Divisors
To find the odd divisors, we need to exclude any factors of 2. Therefore, we only consider the part of the factorization that contains the odd prime factors, which are \(3\) and \(5\).

From the factorization:
\[ 3^3 \times 5^2 \]

### Step 3: Calculate the Number of Odd Divisors
The formula to find the number of divisors from the prime factorization \(p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_n^{e_n}\) is given by:
\[ (e_1 + 1)(e_2 + 1) \ldots (e_n + 1) \]

For our odd part \(3^3 \times 5^2\):
- The exponent of \(3\) is \(3\), so \(e_1 = 3\).
- The exponent of \(5\) is \(2\), so \(e_2 = 2\).

Now, applying the formula:
\[ (3 + 1)(2 + 1) = 4 \times 3 = 12 \]

### Conclusion
Thus, the number of odd divisors of 21600 is **12**.

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