Let X and Y be the sets of all positive divisors of 400 and 1000 respectively (including 1 and the number), Then , `n(X nnY)=`

Options

A. 4
B. 3
C. 8
D. 12

Answer

To find the number of elements in the intersection of the sets \( X \) and \( Y \), we will first determine the positive divisors of 400 and 1000, then find their intersection.

### Step 1: Find the positive divisors of 400
To find the positive divisors of 400, we start with its prime factorization:
\[ 400 = 2^4 \times 5^2 \]
Using the formula for finding the number of divisors, if \( n = p_1^{e_1} \times p_2^{e_2} \), then the number of divisors \( d(n) \) is given by:
\[ d(n) = (e_1 + 1)(e_2 + 1) \]
For 400:
\[ d(400) = (4 + 1)(2 + 1) = 5 \times 3 = 15 \]
Now we list the positive divisors of 400:
- 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400

Thus, the set \( X \) is:
\[ X = \{1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400\} \]

### Step 2: Find the positive divisors of 1000
Next, we find the positive divisors of 1000 by prime factorization:
\[ 1000 = 2^3 \times 5^3 \]
Using the divisor formula:
\[ d(1000) = (3 + 1)(3 + 1) = 4 \times 4 = 16 \]
Now we list the positive divisors of 1000:
- 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000

Thus, the set \( Y \) is:
\[ Y = \{1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000\} \]

### Step 3: Find the intersection of sets \( X \) and \( Y \)
Now we find the intersection \( X \cap Y \):
\[ X \cap Y = \{1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200\} \]

### Step 4: Count the elements in the intersection
Counting the elements in the intersection:
\[ \{1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200\} \text{ has } 12 \text{ elements.} \]

### Final Answer
Thus, the number of elements in \( X \cap Y \) is:
\[ n(X \cap Y) = 12 \]