Let X and Y be the sets of all positive divisions of400 and 1000 respectively (including 1 and the number). Then `n(X cap Y)` is T, then `(T)/(4)` is-

Answer

To solve the problem step by step, we need to find the sets of positive divisors for 400 and 1000, determine their intersection, and then calculate \( \frac{T}{4} \).

### Step 1: Find the positive divisors of 400
To find the positive divisors of 400, we can start by determining its prime factorization.

1. **Prime Factorization of 400**:
\[ 400 = 2^4 \times 5^2 \]

2. **Finding Divisors**:
The formula for finding the number of divisors from the prime factorization \( p_1^{e_1} \times p_2^{e_2} \) is \( (e_1 + 1)(e_2 + 1) \).
- For \( 400 = 2^4 \times 5^2 \):
\[ \text{Number of divisors} = (4 + 1)(2 + 1) = 5 \times 3 = 15 \]

3. **List of Divisors**:
The positive divisors of 400 are:
\[ 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400 \]

### Step 2: Find the positive divisors of 1000
Next, we will find the positive divisors of 1000.

1. **Prime Factorization of 1000**:
\[ 1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3 \]

2. **Finding Divisors**:
- For \( 1000 = 2^3 \times 5^3 \):
\[ \text{Number of divisors} = (3 + 1)(3 + 1) = 4 \times 4 = 16 \]

3. **List of Divisors**:
The positive divisors of 1000 are:
\[ 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000 \]

### Step 3: Find the intersection of sets X and Y
Now, we will find the intersection of the two sets of divisors.

1. **Divisors of 400 (Set X)**:
\[ X = \{1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400\} \]

2. **Divisors of 1000 (Set Y)**:
\[ Y = \{1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000\} \]

3. **Finding Intersection \( X \cap Y \)**:
The common elements in both sets are:
\[ X \cap Y = \{1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200\} \]

### Step 4: Count the number of elements in the intersection
The number of elements in \( X \cap Y \) is:
\[ T = 12 \]

### Step 5: Calculate \( \frac{T}{4} \)
Now we will calculate \( \frac{T}{4} \):
\[ \frac{T}{4} = \frac{12}{4} = 3 \]

### Final Answer
Thus, the final answer is:
\[ \frac{T}{4} = 3 \]