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 $x$ and $y$.
We know that: Product of two numbers = $\text{LCM}\times \text{HCF}$

$$x\cdot y=495\times 5=2475$$

Given: $x+y=100$

Using the algebraic identity:

$$(x-y{)}^2=(x+y{)}^2-4xy$$

$$(x-y{)}^2=(100{)}^2-4(2475)$$

$$(x-y{)}^2=10000-9900=100$$

$$x-y=\sqrt{100}=10$$

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