Find the LCM of : 30, 126 and 180

Answer

To find the LCM (Least Common Multiple) of the numbers 30, 126, and 180, we can use the prime factorization method. Here’s how we can do it step by step:

### Step 1: Prime Factorization of Each Number
1. **Prime Factorization of 30**:
- 30 = 2 × 3 × 5

2. **Prime Factorization of 126**:
- 126 = 2 × 3^2 × 7

3. **Prime Factorization of 180**:
- 180 = 2^2 × 3^2 × 5

### Step 2: Identify the Highest Power of Each Prime Factor
Now, we will identify the highest power of each prime factor from the factorizations:
- For the prime factor **2**: The highest power is 2^2 (from 180).
- For the prime factor **3**: The highest power is 3^2 (from both 126 and 180).
- For the prime factor **5**: The highest power is 5^1 (from both 30 and 180).
- For the prime factor **7**: The highest power is 7^1 (from 126).

### Step 3: Multiply the Highest Powers Together
Now, we will multiply these highest powers together to find the LCM:
- LCM = 2^2 × 3^2 × 5^1 × 7^1

Calculating this:
- 2^2 = 4
- 3^2 = 9
- 5^1 = 5
- 7^1 = 7

Now, multiply these together:
- LCM = 4 × 9 × 5 × 7

### Step 4: Perform the Multiplication
1. First, multiply 4 and 9:
- 4 × 9 = 36
2. Next, multiply the result by 5:
- 36 × 5 = 180
3. Finally, multiply by 7:
- 180 × 7 = 1260

### Conclusion
Thus, the LCM of 30, 126, and 180 is **1260**.

---