Find the even factors of 36.

Answer

To find the even factors of 36, we will follow these steps:

### Step 1: Prime Factorization of 36
First, we need to perform the prime factorization of 36.

- Divide 36 by 2 (the smallest prime number):
- \( 36 \div 2 = 18 \)

- Divide 18 by 2:
- \( 18 \div 2 = 9 \)

- Now, divide 9 by 3 (the next prime number):
- \( 9 \div 3 = 3 \)

- Finally, divide 3 by 3:
- \( 3 \div 3 = 1 \)

So, the prime factorization of 36 is:
\[ 36 = 2^2 \times 3^2 \]

### Step 2: Identify the Even Factors
Even factors are those factors that include at least one factor of 2.

### Step 3: Calculate the Total Number of Factors
The formula to find the total number of factors from the prime factorization \( p_1^{e_1} \times p_2^{e_2} \) is:
\[ (e_1 + 1)(e_2 + 1) \]

For 36:
- The power of 2 is 2 (from \( 2^2 \))
- The power of 3 is 2 (from \( 3^2 \))

So, the total number of factors is:
\[ (2 + 1)(2 + 1) = 3 \times 3 = 9 \]

### Step 4: Calculate the Number of Odd Factors
Odd factors are those factors that do not include 2. We can find the odd factors by considering only the prime factor 3.

Using the same formula:
- The power of 3 is 2.

So, the total number of odd factors is:
\[ (2 + 1) = 3 \]

### Step 5: Calculate the Number of Even Factors
To find the number of even factors, we subtract the number of odd factors from the total number of factors:
\[ \text{Number of even factors} = \text{Total factors} - \text{Odd factors} \]
\[ \text{Number of even factors} = 9 - 3 = 6 \]

### Conclusion
Thus, the number of even factors of 36 is **6**.