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

Let the two numbers be $a$ and $b$.
We know the fundamental property of numbers:

$$\text{Product of two numbers}=\text{LCM}\times \text{HCF}$$

$$a\times b=495\times 5=2475$$

Given the sum of numbers:

$$a+b=100$$

Using the algebraic identity $(a-b{)}^2=(a+b{)}^2-4ab$:

$$(a-b{)}^2=(100{)}^2-4(2475)$$

$$(a-b{)}^2=10000-9900$$

$$(a-b{)}^2=100$$

$$a-b=\sqrt{100}=10$$

Thus, the difference between the two numbers is $10$.