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 find the prime factors of 600, we can divide it by the smallest prime numbers:

- 600 ÷ 2 = 300
- 300 ÷ 2 = 150
- 150 ÷ 2 = 75
- 75 ÷ 3 = 25
- 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 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 prime factorization \( 600 = 2^3 \times 3^1 \times 5^2 \):
- For \( 2^3 \): \( 1 + 2 + 2^2 + 2^3 = 1 + 2 + 4 + 8 = 15 \)
- For \( 3^1 \): \( 1 + 3 = 4 \)
- For \( 5^2 \): \( 1 + 5 + 5^2 = 1 + 5 + 25 = 31 \)

### Step 3: Calculate the sum of the factors
Now we can multiply these results together:
\[ \sigma(600) = 15 \times 4 \times 31 \]

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

### Final Answer
The sum of all factors of 600 is \( 1860 \).

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