Find the LCM of 540 and 108 ?

Answer

To find the Least Common Multiple (LCM) of 540 and 108, we will use the prime factorization method. Here’s a step-by-step solution:

### Step 1: Prime Factorization of 540
- Start by dividing 540 by the smallest prime number, which is 2.
\[ 540 \div 2 = 270 \]
- Divide 270 by 2 again.
\[ 270 \div 2 = 135 \]
- Now, 135 cannot be divided by 2, so we move to the next prime number, which is 3.
- Divide 135 by 3.
\[ 135 \div 3 = 45 \]
- Divide 45 by 3 again.
\[ 45 \div 3 = 15 \]
- Divide 15 by 3 again.
\[ 15 \div 3 = 5 \]
- Finally, 5 is a prime number and cannot be divided further.

Thus, the prime factorization of 540 is:
\[ 540 = 2^2 \times 3^3 \times 5^1 \]

### Step 2: Prime Factorization of 108
- Start by dividing 108 by the smallest prime number, which is 2.
\[ 108 \div 2 = 54 \]
- Divide 54 by 2 again.
\[ 54 \div 2 = 27 \]
- Now, 27 cannot be divided by 2, so we move to the next prime number, which is 3.
- Divide 27 by 3.
\[ 27 \div 3 = 9 \]
- Divide 9 by 3 again.
\[ 9 \div 3 = 3 \]
- Finally, divide 3 by 3.
\[ 3 \div 3 = 1 \]

Thus, the prime factorization of 108 is:
\[ 108 = 2^2 \times 3^3 \]

### Step 3: Finding the LCM
- To find the LCM, we take the highest power of each prime number from both factorizations:
- For 2: \(2^2\) (from both)
- For 3: \(3^3\) (from both)
- For 5: \(5^1\) (only from 540)

So, the LCM is:
\[ LCM = 2^2 \times 3^3 \times 5^1 \]

### Step 4: Calculate the LCM
- Now, calculate the LCM:
\[ LCM = 4 \times 27 \times 5 \]
- First, calculate \(4 \times 27 = 108\).
- Then, calculate \(108 \times 5 = 540\).

Thus, the LCM of 540 and 108 is:
\[ \text{LCM} = 540 \]

### Final Answer
The LCM of 540 and 108 is **540**.

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