The total number of all proper factors of 75600, is

Options

A. 120
B. 119
C. 118
D. none of these

Answer

To find the total number of all proper factors of the number 75600, we will follow these steps:

### Step 1: Prime Factorization of 75600
First, we need to perform the prime factorization of 75600.

- Start dividing by the smallest prime number, which is 2:
- 75600 ÷ 2 = 37800
- 37800 ÷ 2 = 18900
- 18900 ÷ 2 = 9450
- 9450 ÷ 2 = 4725 (we can no longer divide by 2)

- Now divide by the next smallest prime, which is 3:
- 4725 ÷ 3 = 1575
- 1575 ÷ 3 = 525
- 525 ÷ 3 = 175 (we can no longer divide by 3)

- Now divide by the next smallest prime, which is 5:
- 175 ÷ 5 = 35
- 35 ÷ 5 = 7 (we can no longer divide by 5)

- Finally, we have 7, which is a prime number.

So, the prime factorization of 75600 is:
\[ 75600 = 2^4 \times 3^3 \times 5^2 \times 7^1 \]

### Step 2: Use the Formula for Total Factors
The formula to find the total number of factors of a number based on its prime factorization \( n = p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k} \) is:
\[ \text{Total Factors} = (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots (e_k + 1) \]

For our prime factorization \( 2^4 \times 3^3 \times 5^2 \times 7^1 \):
- \( e_1 = 4 \) (for 2)
- \( e_2 = 3 \) (for 3)
- \( e_3 = 2 \) (for 5)
- \( e_4 = 1 \) (for 7)

Now plug these values into the formula:
\[ \text{Total Factors} = (4 + 1)(3 + 1)(2 + 1)(1 + 1) = 5 \times 4 \times 3 \times 2 \]

### Step 3: Calculate Total Factors
Now, calculate the product:
\[ 5 \times 4 = 20 \]
\[ 20 \times 3 = 60 \]
\[ 60 \times 2 = 120 \]

So, the total number of factors of 75600 is 120.

### Step 4: Calculate Proper Factors
Proper factors are all factors excluding the number itself and 1. Therefore, we subtract 2 from the total number of factors:
\[ \text{Proper Factors} = \text{Total Factors} - 2 = 120 - 2 = 118 \]

### Final Answer
The total number of all proper factors of 75600 is **118**.

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