Product of all the even divisors of `N = 1000`, is

Options

A. `2^(20)*5^(20)`
B. `2^(24)*5^(24)`
C. `64*10^(18)`
D. None of these

Explanation

To find the product of all the even divisors of \( N = 1000 \), we will follow these steps:

### Step 1: Factorize \( N \)
First, we need to factor \( N \):
\[ N = 1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3 \]

### Step 2: Identify the even divisors
Even divisors of \( N \) must include at least one factor of 2. The general form of the divisors of \( N \) can be expressed as:
\[ d = 2^a \times 5^b \]
where \( 1 \leq a \leq 3 \) (since we need at least one factor of 2 for even divisors) and \( 0 \leq b \leq 3 \).

### Step 3: List the possible values for \( a \) and \( b \)
The possible values for \( a \) (the power of 2) are:
- \( a = 1 \)
- \( a = 2 \)
- \( a = 3 \)

The possible values for \( b \) (the power of 5) are:
- \( b = 0 \)
- \( b = 1 \)
- \( b = 2 \)
- \( b = 3 \)

### Step 4: Count the even divisors
The total number of even divisors can be calculated as:
\[ \text{Number of choices for } a = 3 \quad (\text{1, 2, or 3}) \]
\[ \text{Number of choices for } b = 4 \quad (\text{0, 1, 2, or 3}) \]
Thus, the total number of even divisors is:
\[ 3 \times 4 = 12 \]

### Step 5: Calculate the product of all even divisors
The product of all divisors of a number \( N \) is given by \( N^{t/2} \), where \( t \) is the total number of divisors.

First, we need to find the total number of divisors of \( N \):
\[ t = (3 + 1)(3 + 1) = 4 \times 4 = 16 \]

Now, the product of all the divisors of \( N \) is:
\[ N^{t/2} = 1000^{16/2} = 1000^8 \]

### Step 6: Find the product of even divisors
Since half of the divisors are even, the product of the even divisors is:
\[ \text{Product of even divisors} = \sqrt{N^{t}} = N^{t/2} \times \sqrt{N^{t/2}} = 1000^{8} \times 1000^{4} = 1000^{12} \]

### Step 7: Simplify \( 1000^{12} \)
Now, we can simplify \( 1000^{12} \):
\[ 1000 = 10^3 \Rightarrow 1000^{12} = (10^3)^{12} = 10^{36} \]

### Final Answer
Thus, the product of all the even divisors of \( N = 1000 \) is:
\[ \boxed{10^{36}} \]