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
We start by finding the prime factorization of 2520.

1. Divide 2520 by 2:
- \( 2520 \div 2 = 1260 \)
2. Divide 1260 by 2:
- \( 1260 \div 2 = 630 \)
3. Divide 630 by 2:
- \( 630 \div 2 = 315 \)
4. Divide 315 by 3:
- \( 315 \div 3 = 105 \)
5. Divide 105 by 3:
- \( 105 \div 3 = 35 \)
6. Divide 35 by 5:
- \( 35 \div 5 = 7 \)
7. Finally, 7 is a prime number.

Thus, 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.

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

Now, substituting the values:
\[ \text{Total Divisors} = (3 + 1)(2 + 1)(1 + 1)(1 + 1) = 4 \times 3 \times 2 \times 2 \]
Calculating this step-by-step:
- \( 4 \times 3 = 12 \)
- \( 12 \times 2 = 24 \)
- \( 24 \times 2 = 48 \)

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

### Step 3: Calculate the Number of Proper Divisors
Proper divisors are calculated by subtracting 2 from the total number of divisors (to exclude 1 and the number itself):
\[ \text{Proper Divisors} = \text{Total Divisors} - 2 = 48 - 2 = 46 \]

### Conclusion
Thus, the number of proper divisors of 2520 is **46**.

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