HCF and LCM: Calculating the Sum of Two Numbers

We are given the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of two numbers. We need to find their sum.

Let the two numbers be \(a\) and \(b\). The fundamental relationship between two numbers, their HCF, and LCM is:

Given:

• \(HCF = 9\)
• \(LCM = 252\)

Using the formula, the product of the two numbers is:

Since the HCF of the numbers is 9, both numbers must be multiples of 9. We can represent the numbers as \(a = 9x\) and \(b = 9y\), where \(x\) and \(y\) are coprime integers (meaning \(HCF(x, y) = 1\)).

Substitute these into the product equation:

Now, solve for the product \(xy\):

We need to find pairs of coprime integers (\(x, y\)) whose product is 28.

The pairs of factors for 28 are (1, 28), (2, 14), and (4, 7).

Let's check which pairs are coprime:

• \(HCF(1, 28) = 1\) (Coprime)
• \(HCF(2, 14) = 2\) (Not coprime)
• \(HCF(4, 7) = 1\) (Coprime)

The coprime pairs \((x, y)\) are (1, 28) and (4, 7).

Now, let's find the numbers and their sums for each coprime pair:

1. Case 1: \((x, y) = (1, 28)\)
   The numbers are \(a = 9 \times 1 = 9\) and \(b = 9 \times 28 = 252\).
   The sum is \(a + b = 9 + 252 = 261\).
2. Case 2: \((x, y) = (4, 7)\)
   The numbers are \(a = 9 \times 4 = 36\) and \(b = 9 \times 7 = 63\).
   The sum is \(a + b = 36 + 63 = 99\).

Comparing these sums with the given options, the sum 99 is present.

Final Answer: The sum of the numbers is 99.