Sum of the factors 48 will be

Options

A. 124
B. 130
C. 132
D. 130

Answer

To find the sum of the factors of the number 48, we can follow these steps:

### Step 1: Prime Factorization of 48
First, we need to factor the number 48 into its prime factors.

- 48 can be divided by 2:
- \( 48 \div 2 = 24 \)
- 24 can be divided by 2:
- \( 24 \div 2 = 12 \)
- 12 can be divided by 2:
- \( 12 \div 2 = 6 \)
- 6 can be divided by 2:
- \( 6 \div 2 = 3 \)
- Finally, 3 is a prime number.

Thus, the prime factorization of 48 is:
\[ 48 = 2^4 \times 3^1 \]

### 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 \( p_1^{k_1} \times p_2^{k_2} \) is:
\[ \text{Sum of factors} = (1 + p_1 + p_1^2 + \ldots + p_1^{k_1})(1 + p_2 + p_2^2 + \ldots + p_2^{k_2}) \]

For our case:
- \( p_1 = 2 \) and \( k_1 = 4 \)
- \( p_2 = 3 \) and \( k_2 = 1 \)

### Step 3: Calculate the Sum for Each Prime Factor
Now we will calculate the sum for each prime factor:

1. For \( 2^4 \):
\[ 1 + 2 + 2^2 + 2^3 + 2^4 = 1 + 2 + 4 + 8 + 16 = 31 \]

2. For \( 3^1 \):
\[ 1 + 3 = 4 \]

### Step 4: Multiply the Results
Now, we multiply the sums obtained from each prime factor:
\[ \text{Sum of factors} = 31 \times 4 = 124 \]

### Final Answer
Thus, the sum of the factors of 48 is **124**.

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