Problem Setup: HCF, LCM, and Difference Given

We are given the Highest Common Factor (HCF) and Least Common Multiple (LCM) of two numbers, along with their difference. We need to find their sum.

• HCF = 12
• LCM = 360
• Difference between the numbers (let them be $a$ and $b$, so $a - b$) = 12

Key Relationship: Product of Numbers

A fundamental property states that the product of two numbers ($a \times b$) is equal to the product of their HCF and LCM.

Calculation:

$a \times b = \text{HCF} \times \text{LCM}$

$a \times b = 12 \times 360$

$a \times b = 4320$

Finding the Sum Using Algebraic Identity

We can use the algebraic identity relating the sum ($a+b$), difference ($a-b$), and product ($ab$) of two numbers:

$(a+b)^2 = (a-b)^2 + 4ab$

Substitute the known values (difference = 12, product $ab = 4320$):

$(a+b)^2 = (12)^2 + 4 \times (4320)$

$(a+b)^2 = 144 + 17280$

$(a+b)^2 = 17424$

Calculating the Sum

To find the sum ($a+b$), take the square root of both sides:

$a+b = \sqrt{17424}$

$a+b = 132$

Thus, the sum of the two numbers is 132.