Finding the Value of n using LCM and HCF

This problem asks us to find the value of an unknown number, represented by 'n', given the Least Common Multiple (LCM) and Highest Common Factor (HCF) of 27 and 'n'. We are provided with the following information:

• The first number is 27.
• The LCM of 27 and 'n' is 54, i.e., \(\text{LCM}(27, n) = 54\).
• The HCF of 27 and 'n' is 9, i.e., \(\text{HCF}(27, n) = 9\).

Understanding the LCM and HCF Relationship

There's a fundamental property that connects two numbers (let's call them 'a' and 'b') with their LCM and HCF. This property states that the product of the two numbers is equal to the product of their LCM and HCF.

Mathematically, this is represented as:

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

Applying the Formula to Find n

In this specific problem, we have:

• \(a = 27\)
• \(b = n\)
• \(\text{HCF}(27, n) = 9\)
• \(\text{LCM}(27, n) = 54\)

Let's substitute these values into the formula:

\(27 \times n = \text{HCF}(27, n) \times \text{LCM}(27, n)\)

\(27 \times n = 9 \times 54\)

Calculating the Value of n

First, calculate the product of the HCF and LCM:

\(9 \times 54 = 486\)

Now, our equation becomes:

\(27 \times n = 486\)

To find 'n', we need to divide 486 by 27:

\(n = \frac{486}{27}\)

Performing the division:

\(n = 18\)

Verification of the Result

Let's check if \(n = 18\) satisfies the given conditions:

• Check HCF(27, 18): The factors of 27 are 1, 3, 9, 27. The factors of 18 are 1, 2, 3, 6, 9, 18. The highest common factor is indeed 9. So, \(\text{HCF}(27, 18) = 9\). This matches the given information.
• Check LCM(27, 18): The multiples of 27 are 27, 54, 81, ... The multiples of 18 are 18, 36, 54, 72, ... The least common multiple is 54. So, \(\text{LCM}(27, 18) = 54\). This also matches the given information.

Since both conditions are met, the calculated value of \(n = 18\) is correct.