How may other factors are there of the number 37800?

Answer

To find the number of other factors of the number 37800, we will follow these steps:

### Step 1: Prime Factorization of 37800
First, we need to perform the prime factorization of the number 37800.

- Divide 37800 by 2:
- 37800 ÷ 2 = 18900
- 18900 ÷ 2 = 9450
- 9450 ÷ 2 = 4725 (no longer divisible by 2)

- Now divide by 3:
- 4725 ÷ 3 = 1575
- 1575 ÷ 3 = 525
- 525 ÷ 3 = 175 (no longer divisible by 3)

- Now divide by 5:
- 175 ÷ 5 = 35
- 35 ÷ 5 = 7 (no longer divisible by 5)

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

Thus, the prime factorization of 37800 is:
\[ 37800 = 2^3 \times 3^3 \times 5^2 \times 7^1 \]

### Step 2: Use the Formula for Total Factors
The formula to find the total number of factors from the prime factorization \( p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \times \ldots \) is:
\[ (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots \]

For our case:
- For \( 2^3 \): \( e_1 = 3 \) → \( 3 + 1 = 4 \)
- For \( 3^3 \): \( e_2 = 3 \) → \( 3 + 1 = 4 \)
- For \( 5^2 \): \( e_3 = 2 \) → \( 2 + 1 = 3 \)
- For \( 7^1 \): \( e_4 = 1 \) → \( 1 + 1 = 2 \)

Now, we can calculate the total number of factors:
\[ \text{Total Factors} = (3 + 1)(3 + 1)(2 + 1)(1 + 1) = 4 \times 4 \times 3 \times 2 \]

### Step 3: Calculate the Total Number of Factors
Calculating the above expression:
\[ 4 \times 4 = 16 \]
\[ 16 \times 3 = 48 \]
\[ 48 \times 2 = 96 \]

So, the total number of factors of 37800 is 96.

### Step 4: Calculate Other Factors
Since we need to find the number of other factors (excluding the number itself), we subtract 1 from the total number of factors:
\[ \text{Other Factors} = 96 - 1 = 95 \]

### Final Answer
Thus, the number of other factors of 37800 is **95**.

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