Sum of factors of 240.

Answer

To find the sum of the factors of 240, we can follow these steps:

### Step 1: Find the prime factorization of 240.
To do this, we divide 240 by the smallest prime numbers until we reach 1.

- 240 ÷ 2 = 120
- 120 ÷ 2 = 60
- 60 ÷ 2 = 30
- 30 ÷ 2 = 15
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1

Thus, the prime factorization of 240 is:
\[ 240 = 2^4 \times 3^1 \times 5^1 \]

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

For \( 240 = 2^4 \times 3^1 \times 5^1 \):
- For \( 2^4 \): \( 1 + 2 + 4 + 8 + 16 = 31 \)
- For \( 3^1 \): \( 1 + 3 = 4 \)
- For \( 5^1 \): \( 1 + 5 = 6 \)

### Step 3: Calculate the sum of the factors.
Now we multiply these sums together:
\[ \sigma(240) = 31 \times 4 \times 6 \]

Calculating step-by-step:
- First, calculate \( 31 \times 4 = 124 \)
- Then, calculate \( 124 \times 6 = 744 \)

### Final Answer:
The sum of the factors of 240 is \( 744 \).

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