The sum of all factors of 600 is

Options

A. 1860
B. 1840
C. 1820
D. 1800

Answer

To find the sum of all factors of 600, we can follow these steps:

### Step 1: Find the prime factorization of 600
To start, we need to break down 600 into its prime factors.

600 can be divided by 2:
- 600 ÷ 2 = 300
- 300 ÷ 2 = 150
- 150 ÷ 2 = 75

Now, 75 is not divisible by 2, so we move to the next prime number, which is 3:
- 75 ÷ 3 = 25

Next, we can divide 25 by 5:
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1

So, the prime factorization of 600 is:
\[ 600 = 2^3 \times 3^1 \times 5^2 \]

### Step 2: Use the formula for the sum of factors
The sum of the factors of a number can be calculated using the formula:
\[ \sigma(n) = (p_1^{k_1 + 1} - 1) / (p_1 - 1) \times (p_2^{k_2 + 1} - 1) / (p_2 - 1) \times \ldots \]
where \( p_1, p_2, \ldots \) are the prime factors and \( k_1, k_2, \ldots \) are their respective powers.

For 600:
- \( p_1 = 2, k_1 = 3 \)
- \( p_2 = 3, k_2 = 1 \)
- \( p_3 = 5, k_3 = 2 \)

### Step 3: Calculate each part of the formula
1. For \( 2^3 \):
\[ \frac{2^{3 + 1} - 1}{2 - 1} = \frac{2^4 - 1}{1} = \frac{16 - 1}{1} = 15 \]

2. For \( 3^1 \):
\[ \frac{3^{1 + 1} - 1}{3 - 1} = \frac{3^2 - 1}{2} = \frac{9 - 1}{2} = \frac{8}{2} = 4 \]

3. For \( 5^2 \):
\[ \frac{5^{2 + 1} - 1}{5 - 1} = \frac{5^3 - 1}{4} = \frac{125 - 1}{4} = \frac{124}{4} = 31 \]

### Step 4: Multiply the results
Now we multiply the results from each part:
\[ \sigma(600) = 15 \times 4 \times 31 \]

Calculating this step-by-step:
1. \( 15 \times 4 = 60 \)
2. \( 60 \times 31 = 1860 \)

### Final Answer
The sum of all factors of 600 is **1860**.

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