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 find the prime factorization of 4200.

- Divide 4200 by 2:
\[ 4200 \div 2 = 2100 \]
- Divide 2100 by 2:
\[ 2100 \div 2 = 1050 \]
- Divide 1050 by 2:
\[ 1050 \div 2 = 525 \]
- Now, 525 is not divisible by 2. Divide by 3:
\[ 525 \div 3 = 175 \]
- Now, 175 is not divisible by 3. Divide by 5:
\[ 175 \div 5 = 35 \]
- Finally, divide 35 by 5:
\[ 35 \div 5 = 7 \]
- 7 is a prime number.

So, the prime factorization of 4200 is:
\[ 4200 = 2^3 \times 3^1 \times 5^2 \times 7^1 \]

### Step 2: Use the Formula for the Number of Divisors
The formula to find the number of divisors \(d(n)\) of a number \(n\) based on its prime factorization \(p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k}\) is:
\[ d(n) = (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots (e_k + 1) \]
Where \(e_i\) are the powers of the prime factors.

### Step 3: Apply the Formula
From the prime factorization \(2^3 \times 3^1 \times 5^2 \times 7^1\):
- The exponent of 2 is 3, so \(e_1 = 3\)
- The exponent of 3 is 1, so \(e_2 = 1\)
- The exponent of 5 is 2, so \(e_3 = 2\)
- The exponent of 7 is 1, so \(e_4 = 1\)

Now, we can substitute these values into the formula:
\[ d(4200) = (3 + 1)(1 + 1)(2 + 1)(1 + 1) \]
Calculating each term:
\[ = 4 \times 2 \times 3 \times 2 \]

### Step 4: Calculate the Result
Now, we compute the product:
\[ = 4 \times 2 = 8 \]
\[ = 8 \times 3 = 24 \]
\[ = 24 \times 2 = 48 \]

Thus, the number of divisors of 4200 is **48**.

### Summary
The number of divisors of 4200 is **48**.

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