Find the LCM of: 42, 30 and 126

Answer

To find the LCM (Least Common Multiple) of the numbers 42, 30, and 126, we can follow these steps:

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

- **For 42**:
- 42 can be divided by 2: \( 42 \div 2 = 21 \)
- 21 can be divided by 3: \( 21 \div 3 = 7 \)
- 7 is a prime number.

So, the prime factorization of 42 is:
\[ 42 = 2^1 \times 3^1 \times 7^1 \]

- **For 30**:
- 30 can be divided by 2: \( 30 \div 2 = 15 \)
- 15 can be divided by 3: \( 15 \div 3 = 5 \)
- 5 is a prime number.

So, the prime factorization of 30 is:
\[ 30 = 2^1 \times 3^1 \times 5^1 \]

- **For 126**:
- 126 can be divided by 2: \( 126 \div 2 = 63 \)
- 63 can be divided by 3: \( 63 \div 3 = 21 \)
- 21 can be divided by 3: \( 21 \div 3 = 7 \)
- 7 is a prime number.

So, the prime factorization of 126 is:
\[ 126 = 2^1 \times 3^2 \times 7^1 \]

### Step 2: Identify the Highest Powers
Next, we identify the highest power of each prime factor from the factorizations:

- For \(2\): The highest power is \(2^1\) (from all three numbers).
- For \(3\): The highest power is \(3^2\) (from 126).
- For \(5\): The highest power is \(5^1\) (from 30).
- For \(7\): The highest power is \(7^1\) (from both 42 and 126).

### Step 3: Calculate the LCM
Now, we multiply these highest powers together to find the LCM:

\[ \text{LCM} = 2^1 \times 3^2 \times 5^1 \times 7^1 \]

Calculating this step-by-step:

1. Calculate \(3^2 = 9\).
2. Multiply \(2 \times 9 = 18\).
3. Multiply \(18 \times 5 = 90\).
4. Finally, multiply \(90 \times 7 = 630\).

Thus, the LCM of 42, 30, and 126 is:
\[ \text{LCM} = 630 \]

### Final Answer
The LCM of 42, 30, and 126 is **630**.

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