The HCF of two numbers of same number of digits is 45 and their LCM is 540. The numbers are

Options

A. 270, 540
B. 135, 270
C. 180, 270
D. 135, 180

Answer

To solve the problem step by step, we will find two numbers whose HCF is 45 and LCM is 540.

### Step 1: Understand the relationship between HCF and LCM
We know that the product of two numbers is equal to the product of their HCF and LCM. This can be expressed as:
\[ \text{Product of the two numbers} = \text{HCF} \times \text{LCM} \]

### Step 2: Substitute the given values
Given:
- HCF = 45
- LCM = 540

Using the formula:
\[ \text{Product of the two numbers} = 45 \times 540 \]

### Step 3: Calculate the product
Now, we calculate the product:
\[ 45 \times 540 = 24300 \]

### Step 4: Express the two numbers in terms of x and y
Let the two numbers be:
- First number = \( 45x \)
- Second number = \( 45y \)

### Step 5: Set up the equation
Using the product we calculated:
\[ 45x \times 45y = 24300 \]
This simplifies to:
\[ 2025xy = 24300 \]

### Step 6: Solve for xy
Now, divide both sides by 2025:
\[ xy = \frac{24300}{2025} \]

Calculating the right side:
\[ xy = 12 \]

### Step 7: Find pairs of (x, y) that satisfy xy = 12
The pairs of integers (x, y) that satisfy \( xy = 12 \) are:
- (1, 12)
- (2, 6)
- (3, 4)
- (4, 3)
- (6, 2)
- (12, 1)

### Step 8: Calculate the corresponding numbers
We will use the pairs (3, 4) and (4, 3) since they will yield the same numbers:
1. For \( (x, y) = (3, 4) \):
- First number = \( 45 \times 3 = 135 \)
- Second number = \( 45 \times 4 = 180 \)

2. For \( (x, y) = (4, 3) \):
- First number = \( 45 \times 4 = 180 \)
- Second number = \( 45 \times 3 = 135 \)

### Step 9: Conclusion
The two numbers are 135 and 180.