Find the LCM of 42, 63 and 231.

Answer

To find the LCM (Least Common Multiple) of the numbers 42, 63, and 231, we will follow these steps:

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

- **Prime Factorization of 42**:
- 42 is divisible by 2: \( 42 \div 2 = 21 \)
- 21 is divisible by 3: \( 21 \div 3 = 7 \)
- 7 is a prime number.

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

- **Prime Factorization of 63**:
- 63 is divisible by 3: \( 63 \div 3 = 21 \)
- 21 is divisible by 3: \( 21 \div 3 = 7 \)
- 7 is a prime number.

So, the prime factorization of 63 is:
\[ 63 = 3^2 \times 7^1 \]

- **Prime Factorization of 231**:
- 231 is divisible by 3: \( 231 \div 3 = 77 \)
- 77 is divisible by 7: \( 77 \div 7 = 11 \)
- 11 is a prime number.

So, the prime factorization of 231 is:
\[ 231 = 3^1 \times 7^1 \times 11^1 \]

### Step 2: Identify Common and Uncommon Factors
Now, we will list the prime factors from all three numbers:
- For 42: \( 2^1, 3^1, 7^1 \)
- For 63: \( 3^2, 7^1 \)
- For 231: \( 3^1, 7^1, 11^1 \)

**Common Factors**:
- \( 3 \) (highest power is \( 3^2 \) from 63)
- \( 7 \) (highest power is \( 7^1 \))

**Uncommon Factors**:
- \( 2 \) (from 42)
- \( 11 \) (from 231)

### Step 3: Calculate the LCM
To find the LCM, we take the highest power of each prime factor:
- From \( 3 \): \( 3^2 \)
- From \( 7 \): \( 7^1 \)
- From \( 2 \): \( 2^1 \)
- From \( 11 \): \( 11^1 \)

Now we multiply these together:
\[ \text{LCM} = 3^2 \times 7^1 \times 2^1 \times 11^1 \]

Calculating this step-by-step:
1. Calculate \( 3^2 = 9 \)
2. Multiply \( 9 \times 7 = 63 \)
3. Multiply \( 63 \times 2 = 126 \)
4. Multiply \( 126 \times 11 = 1386 \)

Thus, the LCM of 42, 63, and 231 is:
\[ \text{LCM} = 1386 \]

### Final Answer
The LCM of 42, 63, and 231 is **1386**.

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