Find the LCM of the numbers given below :
36, 60, 72

Answer

To find the LCM (Least Common Multiple) of the numbers 36, 60, and 72, we can follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of each number.

- **For 36**:
- Divide by 2: \( 36 \div 2 = 18 \)
- Divide by 2: \( 18 \div 2 = 9 \)
- Divide by 3: \( 9 \div 3 = 3 \)
- Divide by 3: \( 3 \div 3 = 1 \)

So, the prime factorization of 36 is \( 2^2 \times 3^2 \).

- **For 60**:
- Divide by 2: \( 60 \div 2 = 30 \)
- Divide by 2: \( 30 \div 2 = 15 \)
- Divide by 3: \( 15 \div 3 = 5 \)
- Divide by 5: \( 5 \div 5 = 1 \)

So, the prime factorization of 60 is \( 2^2 \times 3^1 \times 5^1 \).

- **For 72**:
- Divide by 2: \( 72 \div 2 = 36 \)
- Divide by 2: \( 36 \div 2 = 18 \)
- Divide by 2: \( 18 \div 2 = 9 \)
- Divide by 3: \( 9 \div 3 = 3 \)
- Divide by 3: \( 3 \div 3 = 1 \)

So, the prime factorization of 72 is \( 2^3 \times 3^2 \).

### Step 2: Identify the Highest Powers
Next, we identify the highest powers of all prime factors from the factorizations:

- For the prime number 2: The highest power is \( 2^3 \) (from 72).
- For the prime number 3: The highest power is \( 3^2 \) (from both 36 and 72).
- For the prime number 5: The highest power is \( 5^1 \) (from 60).

### Step 3: Calculate the LCM
Now, we can calculate the LCM by multiplying these highest powers together:

\[ \text{LCM} = 2^3 \times 3^2 \times 5^1 \]

Calculating this step-by-step:

1. \( 2^3 = 8 \)
2. \( 3^2 = 9 \)
3. \( 5^1 = 5 \)

Now, multiply these together:

\[ 8 \times 9 = 72 \]
\[ 72 \times 5 = 360 \]

Thus, the LCM of 36, 60, and 72 is **360**.