The sum of LCM and HCF of two numbers is 8340. If the LCM of these numbers is 8300 more than their HCF, then find the product of the two numbers.

Options

A. 147200
B. 166400
C. 264000
D. 146480

Answer

To solve the problem, let's denote the HCF of the two numbers as \( H \) and the LCM as \( L \).

Given:
1. The sum of LCM and HCF is 8340:
\[ L + H = 8340 \quad \text{(1)} \]
2. The LCM is 8300 more than the HCF:
\[ L = H + 8300 \quad \text{(2)} \]

### Step 1: Substitute Equation (2) into Equation (1)

We can substitute the expression for \( L \) from Equation (2) into Equation (1):
\[ (H + 8300) + H = 8340 \]

### Step 2: Simplify the Equation

Combine like terms:
\[ 2H + 8300 = 8340 \]

### Step 3: Solve for H

Now, isolate \( H \):
\[ 2H = 8340 - 8300 \]
\[ 2H = 40 \]
\[ H = 20 \]

### Step 4: Find L using H

Now that we have \( H \), we can find \( L \) using Equation (2):
\[ L = H + 8300 = 20 + 8300 = 8320 \]

### Step 5: Find the Product of the Two Numbers

The product of two numbers can be calculated using the relationship between LCM and HCF:
\[ \text{Product of the two numbers} = L \times H \]
Substituting the values of \( L \) and \( H \):
\[ \text{Product} = 8320 \times 20 \]

### Step 6: Calculate the Product

Now, perform the multiplication:
\[ \text{Product} = 166400 \]

Thus, the product of the two numbers is \( 166400 \).

### Final Answer:
The product of the two numbers is \( 166400 \).

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