What is the difference between two numbers if their LCM is 495, their HCF is 5, and the sum of the numbers is 100?

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To find the difference between two numbers given that their LCM (Least Common Multiple) is 495, their HCF (Highest Common Factor) is 5, and the sum of the numbers is 100, we can follow these steps:

1. **Understanding the Relationship**: The relationship between the LCM, HCF, and the two numbers (let’s denote them as x and y) is given by the formula:

LCM(x, y) × HCF(x, y) = x × y

2. **Calculating the Product**: Given that:

  • LCM = 495
  • HCF = 5

We can calculate:

x × y = 495 × 5 = 2475

3. **Setting Up Equations**: We also know the sum of the two numbers:

x + y = 100

Now we have two equations:

  • x + y = 100
  • xy = 2475

4. **Substitute y**: We can express y in terms of x:

y = 100 – x

5. **Substitute in the Product Equation**:

Substituting y in the product equation:

x(100 - x) = 2475

This simplifies to:

-x² + 100x - 2475 = 0

6. **Rearranging into a Standard Quadratic Form**:

x² - 100x + 2475 = 0

7. **Solving the Quadratic Equation**: Using the quadratic formula:

x = rac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}

where a = 1, b = -100, c = 2475, we find the discriminant:

Discriminant = b² - 4ac = (-100)² - 4 × 1 × 2475 = 10000 - 9900 = 100

Now substituting back into the formula:

x = rac{100 \\pm \\sqrt{100}}{2} = rac{100 \\pm 10}{2}

This yields two potential solutions for x:

  • x = rac{110}{2} = 55
  • x = rac{90}{2} = 45

8. **Finding y**: Now, plugging these x values back into y = 100 – x :

  • If x = 55, then y = 100 – 55 = 45
  • If x = 45, then y = 100 – 45 = 55

Thus, our two numbers are 55 and 45.

9. **Finding the Difference**: Finally, we can calculate the difference:

Difference = |x - y| = |55 - 45| = 10

Therefore, the difference between the two numbers is 10.

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