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.

216 can be divided by 2:
- 216 ÷ 2 = 108
- 108 ÷ 2 = 54
- 54 ÷ 2 = 27 (27 is not divisible by 2 anymore)

Now, we factor 27:
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

So, 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 factor of 3 from the prime factorization.

### Step 3: Count the Odd Divisors
The odd divisors of 216 will be of the form \( 3^k \), where \( k \) can take values from 0 to the highest power of 3 in the factorization.

From the factorization \( 3^3 \), the possible values for \( k \) are:
- \( k = 0 \) (which gives us \( 3^0 = 1 \))
- \( k = 1 \) (which gives us \( 3^1 = 3 \))
- \( k = 2 \) (which gives us \( 3^2 = 9 \))
- \( k = 3 \) (which gives us \( 3^3 = 27 \))

Thus, \( k \) can take 4 values: 0, 1, 2, and 3.

### Step 4: Calculate the Total Number of Odd Divisors
The total number of odd divisors can be calculated using the formula for the number of divisors based on the prime factorization:
If a number is expressed as \( p_1^{e_1} \times p_2^{e_2} \times ... \times p_n^{e_n} \), the number of divisors is given by:
\[ (e_1 + 1)(e_2 + 1)...(e_n + 1) \]

For our case, since we only have the factor \( 3^3 \):
- The exponent \( e_1 = 3 \)

Thus, the number of odd divisors is:
\[ (3 + 1) = 4 \]

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

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