The number of even proper divisors of 5040, is

Options

A. 48
B. 47
C. 46
D. none of these

Explanation

To find the number of even proper divisors of 5040, we will follow these steps:

### Step 1: Prime Factorization of 5040
First, we need to factor 5040 into its prime factors.

5040 can be expressed as:
\[ 5040 = 2^4 \times 3^2 \times 5^1 \times 7^1 \]

### Step 2: Understanding Even Proper Divisors
Even proper divisors must include at least one factor of 2. Therefore, we will consider the prime factorization while ensuring that the power of 2 is at least 1.

### Step 3: Counting Total Divisors
The formula for finding the total number of divisors from the prime factorization \( p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_n^{e_n} \) is:
\[ (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots (e_n + 1) \]

For 5040:
- For \( 2^4 \): \( 4 + 1 = 5 \)
- For \( 3^2 \): \( 2 + 1 = 3 \)
- For \( 5^1 \): \( 1 + 1 = 2 \)
- For \( 7^1 \): \( 1 + 1 = 2 \)

Thus, the total number of divisors of 5040 is:
\[ 5 \times 3 \times 2 \times 2 = 60 \]

### Step 4: Counting Even Divisors
To find the even divisors, we need to consider the divisors that include at least one factor of 2. We can do this by reducing the power of 2 by 1 (since we need at least one factor of 2).

Now, we consider the prime factorization as:
\[ 2^1 \times 3^2 \times 5^1 \times 7^1 \]

Using the same formula for the number of divisors:
- For \( 2^1 \): \( 1 + 1 = 2 \)
- For \( 3^2 \): \( 2 + 1 = 3 \)
- For \( 5^1 \): \( 1 + 1 = 2 \)
- For \( 7^1 \): \( 1 + 1 = 2 \)

Thus, the total number of even divisors is:
\[ 2 \times 3 \times 2 \times 2 = 24 \]

### Step 5: Counting Proper Divisors
Proper divisors are all divisors excluding the number itself. Since 5040 is included in the total count of 60 divisors, we need to subtract 1 from the total number of even divisors to find the number of even proper divisors.

Thus, the number of even proper divisors is:
\[ 24 - 1 = 23 \]

### Conclusion
The number of even proper divisors of 5040 is **23**.

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