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 find the prime factorization of 5040.

To do this, we can divide 5040 by its smallest prime factors repeatedly:

- 5040 ÷ 2 = 2520
- 2520 ÷ 2 = 1260
- 1260 ÷ 2 = 630
- 630 ÷ 2 = 315
- 315 ÷ 3 = 105
- 105 ÷ 3 = 35
- 35 ÷ 5 = 7
- 7 ÷ 7 = 1

Thus, the prime factorization of 5040 is:
\[ 5040 = 2^4 \times 3^2 \times 5^1 \times 7^1 \]

### Step 2: Total Number of Divisors
To find the total number of divisors of a number given its prime factorization, we use the formula:
\[ \text{Total Divisors} = (e_1 + 1)(e_2 + 1)(e_3 + 1)...(e_n + 1) \]
where \( e_i \) are the exponents in the prime factorization.

For 5040:
- The exponent of 2 is 4, so \( e_1 + 1 = 4 + 1 = 5 \)
- The exponent of 3 is 2, so \( e_2 + 1 = 2 + 1 = 3 \)
- The exponent of 5 is 1, so \( e_3 + 1 = 1 + 1 = 2 \)
- The exponent of 7 is 1, so \( e_4 + 1 = 1 + 1 = 2 \)

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

### Step 3: Finding Even Divisors
To find the number of even divisors, we note that an even divisor must include at least one factor of 2.

We can express the even divisors as:
\[ \text{Even Divisors} = (e_1' + 1)(e_2 + 1)(e_3 + 1)(e_4 + 1) \]
where \( e_1' \) is the exponent of 2 reduced by 1 (since we need at least one factor of 2).

Thus, for even divisors:
- \( e_1' = 4 - 1 = 3 \), so \( e_1' + 1 = 3 + 1 = 4 \)
- \( e_2 + 1 = 3 \)
- \( e_3 + 1 = 2 \)
- \( e_4 + 1 = 2 \)

So, the total number of even divisors is:
\[ 4 \times 3 \times 2 \times 2 = 48 \]

### Step 4: Finding Proper Even Divisors
A proper divisor is any divisor except the number itself. Since 5040 is an even number, we need to exclude it from our count of even divisors.

Thus, the number of even proper divisors is:
\[ \text{Even Proper Divisors} = \text{Even Divisors} - 1 = 48 - 1 = 47 \]

### Final Answer
The number of even proper divisors of 5040 is **47**.

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