Number of even divisions of 504 is

Options

A. 12
B. 24
C. `6.^(3)C_(2)`
D. 18

Answer

To find the number of even divisors of 504, we will follow these steps:

### Step 1: Factorize 504
First, we need to factorize the number 504 into its prime factors.

- Divide 504 by 2:
- \( 504 \div 2 = 252 \)
- Divide 252 by 2:
- \( 252 \div 2 = 126 \)
- Divide 126 by 2:
- \( 126 \div 2 = 63 \)
- Divide 63 by 3:
- \( 63 \div 3 = 21 \)
- Divide 21 by 3:
- \( 21 \div 3 = 7 \)
- Finally, divide 7 by 7:
- \( 7 \div 7 = 1 \)

Thus, the prime factorization of 504 is:
\[ 504 = 2^3 \times 3^2 \times 7^1 \]

### Step 2: Determine the Form of the Divisors
Any divisor of 504 can be expressed in the form:
\[ d = 2^a \times 3^b \times 7^c \]
where:
- \( a \) can take values from 0 to 3 (inclusive),
- \( b \) can take values from 0 to 2 (inclusive),
- \( c \) can take values from 0 to 1 (inclusive).

### Step 3: Identify Conditions for Even Divisors
To find the even divisors, \( a \) must be at least 1 (since an even number must include at least one factor of 2). Therefore, the possible values for \( a \) are:
- \( a = 1, 2, 3 \) (3 options)

### Step 4: Determine the Values for \( b \) and \( c \)
- \( b \) can take values from 0 to 2 (3 options: \( b = 0, 1, 2 \)).
- \( c \) can take values from 0 to 1 (2 options: \( c = 0, 1 \)).

### Step 5: Calculate the Total Number of Even Divisors
Now, we can calculate the total number of even divisors by multiplying the number of options for \( a \), \( b \), and \( c \):
\[ \text{Total Even Divisors} = (\text{Number of options for } a) \times (\text{Number of options for } b) \times (\text{Number of options for } c) \]
\[ = 3 \times 3 \times 2 = 18 \]

### Final Answer
The number of even divisors of 504 is **18**.

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