The number of proper divisors of 2520, is

Options

A. 46
B. 52
C. 64
D. none of these

Explanation

To find the number of proper divisors of 2520, we will follow these steps:

### Step 1: Prime Factorization of 2520
First, we need to find the prime factorization of 2520. We can divide 2520 by prime numbers until we reach 1.

- 2520 ÷ 2 = 1260
- 1260 ÷ 2 = 630
- 630 ÷ 2 = 315
- 315 ÷ 3 = 105
- 105 ÷ 3 = 35
- 35 ÷ 5 = 7
- 7 ÷ 7 = 1

So, the prime factorization of 2520 is:
\[ 2520 = 2^3 \times 3^2 \times 5^1 \times 7^1 \]

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

For 2520:
- The power of 2 is 3, so \( e_1 = 3 \)
- The power of 3 is 2, so \( e_2 = 2 \)
- The power of 5 is 1, so \( e_3 = 1 \)
- The power of 7 is 1, so \( e_4 = 1 \)

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

Calculating this:
\[ 4 \times 3 = 12 \]
\[ 12 \times 2 = 24 \]
\[ 24 \times 2 = 48 \]

So, the total number of divisors of 2520 is 48.

### Step 3: Calculate the Number of Proper Divisors
Proper divisors are all the divisors of a number excluding the number itself. Therefore, to find the number of proper divisors, we subtract 1 from the total number of divisors:
\[ \text{Proper Divisors} = \text{Total Divisors} - 1 = 48 - 1 = 47 \]

### Final Answer
The number of proper divisors of 2520 is 47.

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