The Least Common Multiple and Highest Common Factor of two numbers are 75 and 5 respectively. If their difference is 10, then what will be the sum of these two numbers?

Options

A. 20
B. 25
C. 30
D. 40

Answer

To solve the problem step by step, we will use the information provided about the Least Common Multiple (LCM), Highest Common Factor (HCF), and the difference between the two numbers.

### Step 1: Understand the given information
We are given:
- LCM of two numbers = 75
- HCF of two numbers = 5
- Difference between the two numbers = 10

### Step 2: Use the relationship between LCM, HCF, and the two numbers
The relationship between LCM, HCF, and the two numbers \( x \) and \( y \) can be expressed as:
\[ \text{LCM} \times \text{HCF} = x \times y \]
Substituting the known values:
\[ 75 \times 5 = x \times y \]
Calculating this gives:
\[ 375 = x \times y \]

### Step 3: Set up the equations
We also know from the problem that:
\[ x - y = 10 \]
Now we have two equations:
1. \( x \times y = 375 \)
2. \( x - y = 10 \)

### Step 4: Express one variable in terms of the other
From the second equation, we can express \( x \) in terms of \( y \):
\[ x = y + 10 \]

### Step 5: Substitute into the first equation
Now substitute \( x \) in the first equation:
\[ (y + 10) \times y = 375 \]
Expanding this gives:
\[ y^2 + 10y = 375 \]

### Step 6: Rearrange the equation
Rearranging the equation to set it to zero:
\[ y^2 + 10y - 375 = 0 \]

### Step 7: Solve the quadratic equation
We can solve this quadratic equation using the factorization method. We need two numbers that multiply to \(-375\) and add to \(10\). The numbers are \(25\) and \(-15\):
\[ (y + 25)(y - 15) = 0 \]
Setting each factor to zero gives us:
1. \( y + 25 = 0 \) → \( y = -25 \) (not valid since \( y \) must be positive)
2. \( y - 15 = 0 \) → \( y = 15 \)

### Step 8: Find \( x \)
Now that we have \( y \):
\[ y = 15 \]
Substituting back to find \( x \):
\[ x = y + 10 = 15 + 10 = 25 \]

### Step 9: Calculate the sum of \( x \) and \( y \)
Now we can find the sum of the two numbers:
\[ x + y = 25 + 15 = 40 \]

### Final Answer
The sum of the two numbers is \( 40 \).

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