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

To find the difference between two numbers given their LCM, HCF, and sum, we can use the relationship between these values.

Let the two numbers be a and b. We know from the problem that:

• LCM(a, b) = 495
• HCF(a, b) = 5
• a + b = 100

We also know the relationship between LCM and HCF:

LCM(a, b) × HCF(a, b) = a × b

Substituting the values we have:

495 × 5 = a × b

a × b = 2475

Now we have two equations:

• 1) a + b = 100
• 2) a × b = 2475

Let’s express b in terms of a from the first equation:

b = 100 – a

Now substitute this into the second equation:

a × (100 – a) = 2475

Expanding this, we get:

100a – a² = 2475

Rearranging gives us a quadratic equation:

a² – 100a + 2475 = 0

We can solve this using the quadratic formula:

a = [100 ± √(100² – 4 × 1 × 2475)] / (2 × 1)

a = [100 ± √(10000 – 9900)] / 2

a = [100 ± √100] / 2

a = [100 ± 10] / 2

Calculating the possible values for a:

a = (110) / 2 = 55

a = (90) / 2 = 45

So the two possible pairs are:

• If a = 55, then b = 100 – 55 = 45
• If a = 45, then b = 100 – 45 = 55

Now, we can find the difference:

Difference = |a – b| = |55 – 45| = 10

Thus, the difference between the two numbers is 10.