The LCM of the two numbers is 495 and their HCF is 5. If the sum of the numbers is 100, then find their difference.

Given:

LCM of the two numbers = 495

HCF of the two numbers = 5

Sum of the numbers = 100

Formula Used:

For two numbers a and b:

Product of the numbers = LCM × HCF

Sum of the numbers = a + b

Difference of the numbers = |a - b|

Calculation:

Let the two numbers be a and b.

Given:

a + b = 100

LCM × HCF = a × b

495 × 5 = a × b

⇒ a × b = 2475

We have two equations:

1) a + b = 100

2) a × b = 2475

From equation 1, express b in terms of a:

b = 100 - a

Substitute this into equation 2:

a × (100 - a) = 2475

⇒ 100a - a² = 2475

⇒ a² - 100a + 2475 = 0

Solving the quadratic equation:

\(a = \frac{100 ± √(10000 - 9900)}{2}\)

\(a = \frac{100 ± √(100)}{2}\)

\(a = \frac{100 ± 10}{2}\)

a = (110)/2 or a = (90)/2

a = 55 or a = 45

Therefore, the numbers are 55 and 45.

Their difference:

|55 - 45| = 10

The correct answer is option 2.