The sum and number of even factors of 2450.

Options

A. 3534
B. 183500
C. 123524
D. 42453

Answer

To find the sum and number of even factors of 2450, we can follow these steps:

### Step 1: Prime Factorization of 2450
First, we need to find the prime factorization of 2450.

- Divide 2450 by 2 (the smallest prime number):
\[ 2450 \div 2 = 1225 \]
- Next, divide 1225 by 5:
\[ 1225 \div 5 = 245 \]
- Then divide 245 by 5 again:
\[ 245 \div 5 = 49 \]
- Finally, divide 49 by 7:
\[ 49 \div 7 = 7 \]
- Since 7 is a prime number, we stop here.

Thus, the prime factorization of 2450 is:
\[ 2450 = 2^1 \times 5^2 \times 7^2 \]

### Step 2: Finding Total Number of Factors
To find the total number of factors, we use the formula:
\[ (n_1 + 1)(n_2 + 1)(n_3 + 1) \ldots \]
where \( n_1, n_2, n_3, \ldots \) are the powers of the prime factors.

For 2450:
- The powers are \( 1 \) (for 2), \( 2 \) (for 5), and \( 2 \) (for 7).
- Therefore, the total number of factors is:
\[ (1 + 1)(2 + 1)(2 + 1) = 2 \times 3 \times 3 = 18 \]

### Step 3: Finding Number of Even Factors
Even factors must include the prime factor 2. To find the number of even factors, we can consider the remaining factors (excluding the factor of 2).

The prime factorization for even factors is:
\[ 2^1 \times 5^2 \times 7^2 \]
The number of even factors is calculated by:
\[ (1)(2 + 1)(2 + 1) = 1 \times 3 \times 3 = 9 \]

### Step 4: Finding the Sum of Even Factors
To find the sum of the even factors, we can use the formula for the sum of factors:
\[ \text{Sum} = (p_1^{k_1 + 1} - 1)/(p_1 - 1) \times (p_2^{k_2 + 1} - 1)/(p_2 - 1) \times \ldots \]
For even factors, we consider:
\[ \text{Sum of even factors} = 2 \times \left( (5^0 + 5^1 + 5^2)(7^0 + 7^1 + 7^2) \right) \]
Calculating each part:
- For \( 5 \):
\[ 5^0 + 5^1 + 5^2 = 1 + 5 + 25 = 31 \]
- For \( 7 \):
\[ 7^0 + 7^1 + 7^2 = 1 + 7 + 49 = 57 \]
Now, multiply these results:
\[ 31 \times 57 = 1767 \]
Finally, multiply by 2 (since we need even factors):
\[ \text{Sum of even factors} = 2 \times 1767 = 3534 \]

### Final Results
- The number of even factors of 2450 is **9**.
- The sum of the even factors of 2450 is **3534**.