Finding the Sum of Two Numbers

Given the Highest Common Factor (HCF) and Least Common Multiple (LCM) of two numbers, we can find their sum using properties of numbers.

Given Information

• HCF = 9
• LCM = 252

Using Number Properties

Let the two numbers be \(a\) and \(b\). According to number theory, these numbers can be expressed in terms of their HCF:

• \(a = 9x\)
• \(b = 9y\)

Here, \(x\) and \(y\) must be coprime integers (meaning \(\text{HCF}(x, y) = 1\)).

The LCM of \(a\) and \(b\) is related to \(x\) and \(y\) by the formula:

\(\text{LCM}(a, b) = \text{HCF}(a, b) \times x \times y\)

Substituting the given values:

\(252 = 9 \times x \times y\)

Now, we find the product \(xy\):

\(xy = \frac{252}{9}\)

\(xy = 28\)

Identifying Possible Number Pairs

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

• (1, 28)
• (4, 7)

Let's determine the numbers (\(a, b\)) and their sums for each pair:

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

Conclusion

The calculated possible sums are 261 and 99. The question asks for the sum, and Option D provides the value 90.