The product of the LCM and the HCF of two numbers is 216. The difference between the two numbers is 6. What are the numbers?

Let's assume the two numbers are 'a' and 'b'.

The product of the LCM and the HCF of two numbers is 216.

a $$\times$$ b = 216

ab = 216

$$b=\frac{216}{a}$$    Eq.(i)

The difference between the two numbers is 6.

a-b = 6

Put Eq.(i) in the above equation.

$$a-\frac{216}{a} = 6$$

$$a^2-216=6a$$

$$a^2-6a-216=0$$

$$a^2-(18-12)a-216=0$$

$$a^2-18a+12a-216=0$$

a(a-18)+12(a-18) = 0

(a-18) (a+12) = 0

We know that a negative value is not possible. So a = 18.

Put the value of 'a' in Eq.(i).

$$b=\frac{216}{18}$$

b = 12

So the numbers are 18 and 12.