Two numbers are in ratio 8:9 and their HCF is 6. Their LCM is :

Options

A. 432
B. 54
C. 48
D. 423

Answer

To solve the problem, we need to find the LCM of two numbers that are in the ratio of 8:9 and have a given HCF (or GCD) of 6. Let's break it down step by step:

### Step 1: Define the Numbers
Let the two numbers be \( A \) and \( B \). Since they are in the ratio of 8:9, we can express them in terms of a variable \( x \):
- \( A = 8x \)
- \( B = 9x \)

### Step 2: Use the HCF
We know that the HCF (or GCD) of the two numbers \( A \) and \( B \) is given as 6. The HCF of \( 8x \) and \( 9x \) can be expressed as:
- \( \text{HCF}(8x, 9x) = x \cdot \text{HCF}(8, 9) \)

Since 8 and 9 are co-prime (they have no common factors other than 1), we have:
- \( \text{HCF}(8, 9) = 1 \)

Thus, we can write:
- \( \text{HCF}(8x, 9x) = x \cdot 1 = x \)

Given that the HCF is 6, we can equate:
- \( x = 6 \)

### Step 3: Calculate the Numbers
Now that we have \( x \), we can find the actual numbers:
- \( A = 8x = 8 \times 6 = 48 \)
- \( B = 9x = 9 \times 6 = 54 \)

### Step 4: Calculate the LCM
To find the LCM of two numbers, we can use the relationship between LCM, HCF, and the product of the numbers:
\[ \text{LCM}(A, B) = \frac{A \times B}{\text{HCF}(A, B)} \]

Substituting the values we have:
- \( A = 48 \)
- \( B = 54 \)
- \( \text{HCF}(A, B) = 6 \)

Calculating the product:
\[ A \times B = 48 \times 54 \]

Now, calculate the LCM:
\[ \text{LCM}(A, B) = \frac{48 \times 54}{6} \]

### Step 5: Perform the Calculation
First, calculate \( 48 \times 54 \):
\[ 48 \times 54 = 2592 \]

Now divide by the HCF:
\[ \text{LCM}(A, B) = \frac{2592}{6} = 432 \]

### Final Answer
The LCM of the two numbers is **432**.

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