The number of positive odd divisors of 216 is:

Options

A. 4
B. 6
C. 8
D. 12

Answer

To find the number of positive odd divisors of 216, we can follow these steps:

### Step 1: Prime Factorization of 216
First, we need to find the prime factorization of 216. We can do this by dividing 216 by the smallest prime numbers until we reach 1.

- 216 is even, so we divide by 2:
- \( 216 \div 2 = 108 \)
- \( 108 \div 2 = 54 \)
- \( 54 \div 2 = 27 \)
- Now, 27 is not even, so we divide by the next smallest prime, which is 3:
- \( 27 \div 3 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)

Thus, the prime factorization of 216 is:
\[ 216 = 2^3 \times 3^3 \]

### Step 2: Identify Odd Divisors
Odd divisors are those that do not include the factor of 2. Therefore, we only consider the odd part of the factorization, which is \( 3^3 \).

### Step 3: Calculate the Number of Odd Divisors
To find the number of divisors of a number given its prime factorization, we use the formula:
If \( n = p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k} \), then the number of divisors \( d(n) \) is given by:
\[ d(n) = (e_1 + 1)(e_2 + 1) \ldots (e_k + 1) \]

For the odd part \( 3^3 \):
- The exponent \( e_1 = 3 \).
- Therefore, the number of odd divisors is:
\[ d(3^3) = (3 + 1) = 4 \]

### Conclusion
The number of positive odd divisors of 216 is **4**.

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