Sum of all factors of 248 is

Options

A. 320
B. 480
C. 560
D. 400

Answer

To find the sum of all factors of 248, we can follow these steps:

### Step 1: Find the prime factorization of 248
To find the factors of 248, we first need its prime factorization.

- Start dividing by the smallest prime number, which is 2:
- 248 ÷ 2 = 124
- 124 ÷ 2 = 62
- 62 ÷ 2 = 31 (31 is a prime number)

So, the prime factorization of 248 is:
\[ 248 = 2^3 \times 31^1 \]

### Step 2: Use the formula for the sum of factors
The formula for the sum of the factors of a number based on its prime factorization \( p_1^{k_1} \times p_2^{k_2} \times ... \times p_n^{k_n} \) is:
\[ \sigma(n) = (1 + p_1 + p_1^2 + ... + p_1^{k_1})(1 + p_2 + p_2^2 + ... + p_2^{k_2})...(1 + p_n + p_n^2 + ... + p_n^{k_n}) \]

For our case:
- \( p_1 = 2 \), \( k_1 = 3 \)
- \( p_2 = 31 \), \( k_2 = 1 \)

Using the formula:
\[ \sigma(248) = (1 + 2 + 2^2 + 2^3)(1 + 31) \]

### Step 3: Calculate each part
1. Calculate \( (1 + 2 + 2^2 + 2^3) \):
- \( 1 + 2 + 4 + 8 = 15 \)

2. Calculate \( (1 + 31) \):
- \( 1 + 31 = 32 \)

### Step 4: Multiply the results
Now, multiply the results from Step 3:
\[ \sigma(248) = 15 \times 32 \]

### Step 5: Calculate the final sum
Now calculate \( 15 \times 32 \):
\[ 15 \times 32 = 480 \]

Thus, the sum of all factors of 248 is **480**.

### Final Answer:
The sum of all factors of 248 is **480**.

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