The sum of all the factors of 100 is

Options

A. 115
B. 216
C. 217
D. 223

Answer

To find the sum of all the factors of 100, we can follow these steps:

### Step 1: Prime Factorization of 100
First, we need to express 100 in terms of its prime factors.

100 can be factored as:
\[ 100 = 10 \times 10 = (2 \times 5) \times (2 \times 5) = 2^2 \times 5^2 \]

### Step 2: Use the Formula for the Sum of Factors
The formula for the sum of the factors of a number \( n \) that can be expressed as \( p_1^{k_1} \times p_2^{k_2} \times ... \times p_m^{k_m} \) is:
\[ \sigma(n) = (1 + p_1 + p_1^2 + ... + p_1^{k_1})(1 + p_2 + p_2^2 + ... + p_2^{k_2}) \ldots (1 + p_m + p_m^2 + ... + p_m^{k_m}) \]

For our case, \( n = 100 = 2^2 \times 5^2 \).

### Step 3: Calculate Each Part of the Formula
Now, we will apply the formula to our prime factors:

1. For \( 2^2 \):
\[ 1 + 2 + 4 = 7 \]

2. For \( 5^2 \):
\[ 1 + 5 + 25 = 31 \]

### Step 4: Multiply the Results
Now we multiply the results from the previous step:
\[ \sigma(100) = (1 + 2 + 4)(1 + 5 + 25) = 7 \times 31 \]

### Step 5: Calculate the Final Result
Now we calculate \( 7 \times 31 \):
\[ 7 \times 31 = 217 \]

### Conclusion
Thus, the sum of all the factors of 100 is:
\[ \boxed{217} \]