The number of divisor of 4200, is

Options

A. 42
B. 48
C. 54
D. none of these

Explanation

To find the number of divisors of the number 4200, we will follow these steps:

### Step 1: Prime Factorization of 4200
First, we need to perform the prime factorization of 4200.

1. Divide 4200 by 10 (which is 2 × 5):
\[ 4200 \div 10 = 420 \]
2. Divide 420 by 10:
\[ 420 \div 10 = 42 \]
3. Now, factor 42:
\[ 42 = 2 \times 21 = 2 \times 3 \times 7 \]
4. Combining all the factors, we have:
\[ 4200 = 2^3 \times 3^1 \times 5^2 \times 7^1 \]

### Step 2: Using the Divisor Formula
The formula to find the number of divisors from the prime factorization \( p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \) is:
\[ (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots \]
where \( e_1, e_2, e_3, \ldots \) are the powers of the prime factors.

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

### Step 3: Applying the Formula
Now we apply the formula:
\[ (3 + 1)(1 + 1)(2 + 1)(1 + 1) \]
Calculating each term:
- \( 3 + 1 = 4 \)
- \( 1 + 1 = 2 \)
- \( 2 + 1 = 3 \)
- \( 1 + 1 = 2 \)

Now multiply these results:
\[ 4 \times 2 \times 3 \times 2 \]

### Step 4: Performing the Multiplication
Calculating step by step:
1. \( 4 \times 2 = 8 \)
2. \( 8 \times 3 = 24 \)
3. \( 24 \times 2 = 48 \)

### Conclusion
Thus, the total number of divisors of 4200 is:
\[ \boxed{48} \]