There are n rectangles each with area `200 cm^(2)`. If the dimensions of each n rectangles are in integers then the valu of n is :

Options

A. 4
B. 6
C. 12
D. none of these

Answer

To solve the problem of finding the value of \( n \) for the rectangles with an area of \( 200 \, \text{cm}^2 \) and integer dimensions, we can follow these steps:

### Step 1: Understand the Area of a Rectangle
The area \( A \) of a rectangle is given by the formula:
\[ A = \text{Length} \times \text{Breadth} \]
In this case, we know that the area is \( 200 \, \text{cm}^2 \).

### Step 2: Set Up the Equation
We can set up the equation:
\[ \text{Length} \times \text{Breadth} = 200 \]
Let’s denote Length as \( L \) and Breadth as \( B \). Thus, we have:
\[ L \times B = 200 \]

### Step 3: Find Integer Factor Pairs
To find the integer dimensions, we need to determine the pairs of integers \( (L, B) \) that multiply to \( 200 \). We will find all factor pairs of \( 200 \).

### Step 4: Factorization of 200
First, we find the prime factorization of \( 200 \):
\[ 200 = 2^3 \times 5^2 \]

### Step 5: List All Factor Pairs
Now we will find all pairs of factors of \( 200 \):
1. \( 1 \times 200 \)
2. \( 2 \times 100 \)
3. \( 4 \times 50 \)
4. \( 5 \times 40 \)
5. \( 8 \times 25 \)
6. \( 10 \times 20 \)

### Step 6: Count Unique Combinations
Each factor pair \( (L, B) \) can also be reversed to give another valid rectangle \( (B, L) \). However, since \( L \) and \( B \) are interchangeable, we will count each unique pair only once.

The unique pairs are:
1. \( (1, 200) \)
2. \( (2, 100) \)
3. \( (4, 50) \)
4. \( (5, 40) \)
5. \( (8, 25) \)
6. \( (10, 20) \)

### Step 7: Total Combinations
Counting all unique combinations, we find:
- \( (1, 200) \)
- \( (2, 100) \)
- \( (4, 50) \)
- \( (5, 40) \)
- \( (8, 25) \)
- \( (10, 20) \)

Including the reverse pairs, we have:
1. \( (1, 200) \)
2. \( (200, 1) \)
3. \( (2, 100) \)
4. \( (100, 2) \)
5. \( (4, 50) \)
6. \( (50, 4) \)
7. \( (5, 40) \)
8. \( (40, 5) \)
9. \( (8, 25) \)
10. \( (25, 8) \)
11. \( (10, 20) \)
12. \( (20, 10) \)

Thus, we have a total of \( 12 \) unique combinations.

### Conclusion
The value of \( n \) is \( 12 \).