Find the LCM of 1.8, 0.54 and 7.2.

Answer

To find the LCM (Least Common Multiple) of the numbers 1.8, 0.54, and 7.2, we can follow these steps:

### Step 1: Convert the decimal numbers to fractions.
- 1.8 can be written as \( \frac{18}{10} \)
- 0.54 can be written as \( \frac{54}{100} \)
- 7.2 can be written as \( \frac{72}{10} \)

### Step 2: Identify the numerators and denominators.
- The numerators are: 18, 54, and 72.
- The denominators are: 10, 100, and 10.

### Step 3: Find the LCM of the numerators.
To find the LCM of 18, 54, and 72:
- The prime factorization of each number is:
- 18 = \( 2 \times 3^2 \)
- 54 = \( 2 \times 3^3 \)
- 72 = \( 2^3 \times 3^2 \)

- The LCM is found by taking the highest power of each prime factor:
- For 2: the highest power is \( 2^3 \)
- For 3: the highest power is \( 3^3 \)

So, the LCM of the numerators is:
\[ LCM = 2^3 \times 3^3 = 8 \times 27 = 216 \]

### Step 4: Find the GCD (Greatest Common Divisor) of the denominators.
To find the GCD of 10, 100, and 10:
- The prime factorization is:
- 10 = \( 2 \times 5 \)
- 100 = \( 2^2 \times 5^2 \)

- The GCD is found by taking the lowest power of each prime factor:
- For 2: the lowest power is \( 2^1 \)
- For 5: the lowest power is \( 5^0 \) (since 10 appears as \( 5^1 \))

So, the GCD of the denominators is:
\[ GCD = 2^1 \times 5^0 = 2 \]

### Step 5: Calculate the LCM of the fractions.
The LCM of the fractions is given by:
\[ LCM = \frac{LCM \text{ of numerators}}{GCD \text{ of denominators}} \]
Substituting the values we found:
\[ LCM = \frac{216}{2} = 108 \]

### Conclusion:
The LCM of 1.8, 0.54, and 7.2 is **108**.

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