Convert Numbers to Fractions

First, represent the given decimal numbers 3.6, 0.009, and 0.27 as fractions. To do this, place the number without the decimal point over the appropriate power of 10.

• \(3.6 = \frac{36}{10}\)
• \(0.009 = \frac{9}{1000}\)
• \(0.27 = \frac{27}{100}\)

Find LCM of Numerators

Next, calculate the Least Common Multiple (LCM) of the numerators of these fractions: 36, 9, and 27.

Use prime factorization for each numerator:

• \(36 = 2^2 \times 3^2\)
• \(9 = 3^2\)
• \(27 = 3^3\)

The LCM is found by taking the highest power of each prime factor present in any of the numbers: \(LCM(36, 9, 27) = 2^2 \times 3^3 = 4 \times 27 = 108\).

Find GCD of Denominators

Then, calculate the Greatest Common Divisor (GCD) of the denominators: 10, 1000, and 100.

Use prime factorization for each denominator:

• \(10 = 2^1 \times 5^1\)
• \(1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3\)
• \(100 = 10^2 = (2 \times 5)^2 = 2^2 \times 5^2\)

The GCD is found by taking the lowest power of each prime factor common to all the numbers: \(GCD(10, 1000, 100) = 2^1 \times 5^1 = 10\).

Compute LCM of Decimal Numbers

The formula for the LCM of fractions \(\frac{a}{b}, \frac{c}{d}, \frac{e}{f}\) is \(\frac{LCM(a, c, e)}{GCD(b, d, f)}\). Apply this formula to find the LCM of the original decimal numbers.

\(LCM(3.6, 0.009, 0.27) = \frac{LCM(\text{Numerators})}{\text{GCD}(\text{Denominators})} = \frac{108}{10}\)

\(LCM = 10.8\).