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 proper factors of 75600, we will follow these steps:

### Step 1: Prime Factorization of 75600
First, we need to factor 75600 into its prime factors.

- Start with 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 as 4725 is odd)

- Next, divide by 3:
- 4725 ÷ 3 = 1575
- 1575 ÷ 3 = 525
- 525 ÷ 3 = 175 (We can no longer divide by 3 as 175 is not divisible by 3)

- Now, divide by 5:
- 175 ÷ 5 = 35
- 35 ÷ 5 = 7 (7 is a prime number)

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

### Step 2: Calculate the Total Number of Factors
To find the total number of factors, we use the formula:
\[ \text{Total Factors} = (e_1 + 1)(e_2 + 1)(e_3 + 1)(e_4 + 1) \]
where \( e_1, e_2, e_3, e_4 \) are the powers of the prime factors.

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

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

Calculating this:
\[ 5 \times 4 = 20 \]
\[ 20 \times 3 = 60 \]
\[ 60 \times 2 = 120 \]

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

### Step 3: Calculate the Number of Proper Factors
Proper factors are all factors excluding the number itself and 1.

To find the number of proper factors:
\[ \text{Proper Factors} = \text{Total Factors} - 2 \]
(where we subtract 1 for the number itself and 1 for the factor 1)

So:
\[ \text{Proper Factors} = 120 - 2 = 118 \]

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

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