Number of proper factors of 2400 is equal to
(a) 34 (b) 35 (c) 36 (d) 37

Options

A. 34
B. 35
C. 36
D. 37

Answer

To find the number of proper factors of the number 2400, we can follow these steps:

### Step 1: Prime Factorization of 2400
First, we need to find the prime factorization of the number 2400.

2400 can be factored as follows:
- 2400 = 24 × 100
- 24 = 2^3 × 3^1
- 100 = 10^2 = (2 × 5)^2 = 2^2 × 5^2

Combining these, we get:
\[ 2400 = 2^3 \times 3^1 \times 2^2 \times 5^2 = 2^{3+2} \times 3^1 \times 5^2 = 2^5 \times 3^1 \times 5^2 \]

### Step 2: Use the Formula for the Number of Factors
To find the total number of factors of a number given its prime factorization \( p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \), we use the formula:
\[ \text{Total Factors} = (e_1 + 1)(e_2 + 1)(e_3 + 1) \]

For 2400, the exponents from the prime factorization \( 2^5 \times 3^1 \times 5^2 \) are:
- \( e_1 = 5 \) (for 2)
- \( e_2 = 1 \) (for 3)
- \( e_3 = 2 \) (for 5)

Now, applying the formula:
\[ \text{Total Factors} = (5 + 1)(1 + 1)(2 + 1) \]
\[ = 6 \times 2 \times 3 \]
\[ = 36 \]

### Step 3: Calculate the Number of Proper Factors
Proper factors of a number are all factors excluding the number itself and 1. Therefore, we need to subtract 2 (for the number itself and 1) from the total number of factors.

So, we calculate:
\[ \text{Proper Factors} = \text{Total Factors} - 2 \]
\[ = 36 - 2 \]
\[ = 34 \]

### Final Answer
Thus, the number of proper factors of 2400 is **34**.