Find the LCM of 420, 9009, 6270.

Answer

To find the LCM (Least Common Multiple) of the numbers 420, 9009, and 6270, we will follow these steps:

### Step 1: Prime Factorization of Each Number

1. **Prime Factorization of 420:**
- Divide by 2: \( 420 \div 2 = 210 \)
- Divide by 2: \( 210 \div 2 = 105 \)
- Divide by 3: \( 105 \div 3 = 35 \)
- Divide by 5: \( 35 \div 5 = 7 \)
- 7 is a prime number.
- So, the prime factorization of 420 is:
\[ 420 = 2^2 \times 3^1 \times 5^1 \times 7^1 \]

2. **Prime Factorization of 9009:**
- Divide by 3: \( 9009 \div 3 = 3003 \)
- Divide by 3: \( 3003 \div 3 = 1001 \)
- Divide by 7: \( 1001 \div 7 = 143 \)
- Divide by 11: \( 143 \div 11 = 13 \)
- 13 is a prime number.
- So, the prime factorization of 9009 is:
\[ 9009 = 3^2 \times 7^1 \times 11^1 \times 13^1 \]

3. **Prime Factorization of 6270:**
- Divide by 2: \( 6270 \div 2 = 3135 \)
- Divide by 3: \( 3135 \div 3 = 1045 \)
- Divide by 5: \( 1045 \div 5 = 209 \)
- 209 is not divisible by 2, 3, or 5, but it is divisible by 11: \( 209 \div 11 = 19 \)
- 19 is a prime number.
- So, the prime factorization of 6270 is:
\[ 6270 = 2^1 \times 3^1 \times 5^1 \times 11^1 \times 19^1 \]

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

Now, we will take the highest power of each prime factor from all three factorizations:

- For \(2\): Highest power is \(2^2\) (from 420)
- For \(3\): Highest power is \(3^2\) (from 9009)
- For \(5\): Highest power is \(5^1\) (from both 420 and 6270)
- For \(7\): Highest power is \(7^1\) (from both 420 and 9009)
- For \(11\): Highest power is \(11^1\) (from 9009 and 6270)
- For \(13\): Highest power is \(13^1\) (from 9009)
- For \(19\): Highest power is \(19^1\) (from 6270)

### Step 3: Calculate the LCM

Now, we multiply these highest powers together to find the LCM:

\[ \text{LCM} = 2^2 \times 3^2 \times 5^1 \times 7^1 \times 11^1 \times 13^1 \times 19^1 \]

Calculating this step by step:

1. \(2^2 = 4\)
2. \(3^2 = 9\)
3. Multiply: \(4 \times 9 = 36\)
4. Multiply: \(36 \times 5 = 180\)
5. Multiply: \(180 \times 7 = 1260\)
6. Multiply: \(1260 \times 11 = 13860\)
7. Multiply: \(13860 \times 13 = 180180\)
8. Multiply: \(180180 \times 19 = 3423420\)

Thus, the LCM of 420, 9009, and 6270 is:

\[ \text{LCM} = 3423420 \]

### Summary of Steps:
1. Prime factorize each number.
2. Identify the highest powers of all prime factors.
3. Multiply these highest powers to find the LCM.

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