Question

Find the LCM by long division method
70, 110 and 150

Answer 1

To find the least common multiple (LCM) of the numbers 70, 110, and 150 using the long division method, follow these steps:

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

- **70:**
- 70 = 2 × 35
- 35 = 5 × 7
- So, \(70 = 2^1 \times 5^1 \times 7^1\)

- **110:**
- 110 = 2 × 55
- 55 = 5 × 11
- So, \(110 = 2^1 \times 5^1 \times 11^1\)

- **150:**
- 150 = 2 × 75
- 75 = 3 × 25
- 25 = 5 × 5
- So, \(150 = 2^1 \times 3^1 \times 5^2\)

### Step 2: Identify Maximum Powers of Each Prime Factor
Next, list the prime factors and take the highest power of each prime that appears in the factorizations:

- For \(2\): The highest power is \(2^1\) (common to all three numbers).
- For \(3\): The highest power is \(3^1\) (appears only in 150).
- For \(5\): The highest power is \(5^2\) (from 150).
- For \(7\): The highest power is \(7^1\) (from 70).
- For \(11\): The highest power is \(11^1\) (from 110).

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

\[
LCM = 2^1 \times 3^1 \times 5^2 \times 7^1 \times 11^1
\]

Calculating this step by step:

\[
= 2 \times 3 \times 25 \times 7 \times 11
\]

Calculating \(2 \times 3 = 6\)

Now, \(6 \times 25 = 150\)

Then, \(150 \times 7 = 1050\)

Finally, \(1050 \times 11 = 11550\)

### Step 4: Conclusion
Thus, the least common multiple (LCM) of 70, 110, and 150 is:

\[
\boxed{11550}
\]