To calculate the least common multiple (LCM), the numbers need to be whole numbers or rational fractions that can be expressed with integer numerators and denominators. Since two of the numbers here (2.7 and 0.09) are decimals, we should first convert them into fractions.

- \( 3 \) is already a whole number.
- \( 2.7 = \frac{27}{10} \)
- \( 0.09 = \frac{9}{100} \)

Now we need to calculate the LCM of \( 3 \), \( \frac{27}{10} \), and \( \frac{9}{100} \).

### Step 1: Convert fractions to have a common denominator
- \( 3 = \frac{3}{1} \)
- \( \frac{27}{10} \)
- \( \frac{9}{100} \)

The denominators are \( 1 \), \( 10 \), and \( 100 \). The least common denominator (LCD) of \( 1 \), \( 10 \), and \( 100 \) is \( 100 \). So, convert all fractions to have a denominator of 100:

- \( 3 = \frac{300}{100} \)
- \( 2.7 = \frac{270}{100} \)
- \( 0.09 = \frac{9}{100} \)

### Step 2: Find the LCM of the numerators
Now, find the LCM of the numerators \( 300 \), \( 270 \), and \( 9 \).

- Prime factors of \( 300 = 2^2 \times 3 \times 5^2 \)
- Prime factors of \( 270 = 2 \times 3^3 \times 5 \)
- Prime factors of \( 9 = 3^2 \)

The LCM is determined by taking the highest powers of all prime factors:
- \( 2^2 \) (from 300)
- \( 3^3 \) (from 270)
- \( 5^2 \) (from 300)

Thus, the LCM of \( 300 \), \( 270 \), and \( 9 \) is:
\[
LCM = 2^2 \times 3^3 \times 5^2 = 4 \times 27 \times 25 = 2700
\]

### Step 3: Convert back to the original form
Since all the fractions had a denominator of 100, the LCM of the original numbers is:
\[
\frac{2700}{100} = 27
\]

Thus, the least common multiple (LCM) of \( 3 \), \( 2.7 \), and \( 0.09 \) is **27**.

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Would you like further explanation or details on this?

Here are five related questions for practice:
1. What is the LCM of \( 5 \), \( 3.5 \), and \( 0.07 \)?
2. How do you find the LCM of two or more fractions?
3. What is the greatest common divisor (GCD) of \( 3 \), \( 2.7 \), and \( 0.09 \)?
4. Can LCM be applied to irrational numbers? Why or why not?
5. How do you calculate the LCM of three whole numbers using prime factorization?

**Tip**: When dealing with decimals in LCM problems, convert them to fractions for easier calculation.

Learn how to calculate the least common multiple (LCM) of 3, 2.7, and 0.09 by converting decimals to fractions and using prime factorization. This step-by-step guide explains how to handle decimals in LCM problems and includes important mathematical concepts like fractions and LCM. Ideal for Grades 8-10 students.

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