Find the LCM of 108, 135 and 162.

Answer

To find the LCM (Least Common Multiple) of the numbers 108, 135, and 162, we will follow these steps:

### Step 1: Find the Prime Factorization of Each Number

1. **Prime Factorization of 108:**
- Start with 108. It can be divided by 2:
\[ 108 \div 2 = 54 \]
- Divide 54 by 2:
\[ 54 \div 2 = 27 \]
- Now, divide 27 by 3:
\[ 27 \div 3 = 9 \]
- Divide 9 by 3:
\[ 9 \div 3 = 3 \]
- Finally, divide 3 by 3:
\[ 3 \div 3 = 1 \]
- Thus, the prime factorization of 108 is:
\[ 108 = 2^2 \times 3^3 \]

2. **Prime Factorization of 135:**
- Start with 135. It can be divided by 3:
\[ 135 \div 3 = 45 \]
- Divide 45 by 3:
\[ 45 \div 3 = 15 \]
- Divide 15 by 3:
\[ 15 \div 3 = 5 \]
- Finally, divide 5 by 5:
\[ 5 \div 5 = 1 \]
- Thus, the prime factorization of 135 is:
\[ 135 = 3^3 \times 5^1 \]

3. **Prime Factorization of 162:**
- Start with 162. It can be divided by 2:
\[ 162 \div 2 = 81 \]
- Divide 81 by 3:
\[ 81 \div 3 = 27 \]
- Divide 27 by 3:
\[ 27 \div 3 = 9 \]
- Divide 9 by 3:
\[ 9 \div 3 = 3 \]
- Finally, divide 3 by 3:
\[ 3 \div 3 = 1 \]
- Thus, the prime factorization of 162 is:
\[ 162 = 2^1 \times 3^4 \]

### Step 2: Identify the Highest Powers of Each Prime Factor

Now we will find the highest power of each prime factor from the factorizations:
- For \(2\): The highest power is \(2^2\) (from 108).
- For \(3\): The highest power is \(3^4\) (from 162).
- For \(5\): The highest power is \(5^1\) (from 135).

### Step 3: Calculate the LCM

The LCM is found by multiplying the highest powers of all prime factors:
\[ \text{LCM} = 2^2 \times 3^4 \times 5^1 \]

Calculating this step-by-step:
1. Calculate \(2^2 = 4\).
2. Calculate \(3^4 = 81\).
3. Calculate \(5^1 = 5\).

Now, multiply these results together:
\[ \text{LCM} = 4 \times 81 \times 5 \]

Calculating \(4 \times 81\):
\[ 4 \times 81 = 324 \]
Then, calculate \(324 \times 5\):
\[ 324 \times 5 = 1620 \]

Thus, the LCM of 108, 135, and 162 is:
\[ \text{LCM} = 1620 \]