What is the LCM of 84, 140 and 70?

Options

A. 210
B. 420
C. 630
D. 294

Answer

To find the LCM (Least Common Multiple) of the numbers 84, 140, and 70, we can follow these steps:

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

- **For 84**:
- Divide by 2: \(84 \div 2 = 42\)
- Divide by 2: \(42 \div 2 = 21\)
- Divide by 3: \(21 \div 3 = 7\)
- Divide by 7: \(7 \div 7 = 1\)

So, the prime factorization of 84 is \(2^2 \times 3^1 \times 7^1\).

- **For 140**:
- Divide by 2: \(140 \div 2 = 70\)
- Divide by 2: \(70 \div 2 = 35\)
- Divide by 5: \(35 \div 5 = 7\)
- Divide by 7: \(7 \div 7 = 1\)

So, the prime factorization of 140 is \(2^2 \times 5^1 \times 7^1\).

- **For 70**:
- Divide by 2: \(70 \div 2 = 35\)
- Divide by 5: \(35 \div 5 = 7\)
- Divide by 7: \(7 \div 7 = 1\)

So, the prime factorization of 70 is \(2^1 \times 5^1 \times 7^1\).

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

- For \(2\): The highest power is \(2^2\) (from 84 and 140).
- For \(3\): The highest power is \(3^1\) (from 84).
- For \(5\): The highest power is \(5^1\) (from 140 and 70).
- For \(7\): The highest power is \(7^1\) (from all three numbers).

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

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

Calculating this step by step:

1. \(2^2 = 4\)
2. \(4 \times 3 = 12\)
3. \(12 \times 5 = 60\)
4. \(60 \times 7 = 420\)

Thus, the LCM of 84, 140, and 70 is **420**.