HCF of three numbers is 3 and LCM is 27. Find the product of the three numbers.

1. Break Down the Problem

We need to find the product of three numbers given their Highest Common Factor (HCF) and Least Common Multiple (LCM). The HCF of the three numbers is 3 and the LCM is 27.

2. Relevant Concepts

The relationship between the product of the numbers, their HCF, and their LCM is given by the formula:

$\text{Product of the numbers} = \text{HCF} \times \text{LCM}^n$

where $n$ is the number of numbers. For three numbers, $n = 3$.

3. Analysis and Detail

Substituting the known values into the formula:

• HCF = 3
• LCM = 27
• $n = 3$

Thus, the product of the three numbers can be calculated as:

$\text{Product} = 3 \times 27^3$

Calculating $27^3$:

$27^3 = 27 \times 27 \times 27 = 729 \times 27 = 19683$

Now, substituting back to find the product:

$\text{Product} = 3 \times 19683$

Calculating $3 \times 19683$:

$3 \times 19683 = 59049$

4. Verify and Summarize

Thus, the calculations confirm that the product of the three numbers is 59049, and it consists of the HCF and the LCM correctly applied.

Final Answer

The product of the three numbers is 59049.