Find the number of even factors of 60060.

Options

A. a) 128
B. b) 64
C. c) 32
D. d) 80

Answer

To find the number of even factors of 60060, we will follow these steps:

### Step 1: Prime Factorization of 60060
First, we need to find the prime factorization of 60060.

1. **Divide by 2 (the smallest prime number)**:
- 60060 ÷ 2 = 30030
- 30030 ÷ 2 = 15015 (2 appears twice)

2. **Next, divide by 3**:
- 15015 ÷ 3 = 5005 (3 appears once)

3. **Next, divide by 5**:
- 5005 ÷ 5 = 1001 (5 appears once)

4. **Next, divide by 7**:
- 1001 ÷ 7 = 143 (7 appears once)

5. **Next, divide by 11**:
- 143 ÷ 11 = 13 (11 appears once)

6. **Finally, divide by 13**:
- 13 ÷ 13 = 1 (13 appears once)

So, the prime factorization of 60060 is:
\[ 60060 = 2^2 \times 3^1 \times 5^1 \times 7^1 \times 11^1 \times 13^1 \]

### Step 2: Finding the Total Number of Factors
To find the total number of factors, we use the formula:
\[ (e_1 + 1)(e_2 + 1)(e_3 + 1)...(e_n + 1) \]
where \( e_i \) are the powers of the prime factors.

From our factorization:
- The powers are: \( 2^2, 3^1, 5^1, 7^1, 11^1, 13^1 \)

Calculating the total number of factors:
\[ (2 + 1)(1 + 1)(1 + 1)(1 + 1)(1 + 1)(1 + 1) \]
\[ = 3 \times 2 \times 2 \times 2 \times 2 \times 2 \]
\[ = 3 \times 2^5 \]
\[ = 3 \times 32 = 96 \]

### Step 3: Finding the Number of Even Factors
To find the number of even factors, we note that an even factor must include at least one factor of 2.

1. **Consider the prime factorization without the factor of 2**:
- The remaining factors are: \( 3^1, 5^1, 7^1, 11^1, 13^1 \)

2. **Calculating the number of factors of the remaining part**:
\[ (1 + 1)(1 + 1)(1 + 1)(1 + 1)(1 + 1) \]
\[ = 2 \times 2 \times 2 \times 2 \times 2 \]
\[ = 2^5 = 32 \]

3. **Since each of these factors can be multiplied by the factors of \( 2^2 \)**:
- The number of even factors is:
\[ 32 \times 2 = 64 \]

### Final Answer
The number of even factors of 60060 is **64**.

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