Using prime factorisation, find the LCM of 150 and 210.

Answer

To find the LCM (Lowest Common Multiple) of 150 and 210 using prime factorization, we will follow these steps:

### Step 1: Prime Factorization of 150
First, we need to find the prime factors of 150.
- Start dividing by the smallest prime number, which is 2:
- 150 ÷ 2 = 75
- Next, divide 75 by the next smallest prime number, which is 3:
- 75 ÷ 3 = 25
- Now, divide 25 by the next smallest prime number, which is 5:
- 25 ÷ 5 = 5
- Finally, divide 5 by itself:
- 5 ÷ 5 = 1

So, the prime factorization of 150 is:
\[ 150 = 2^1 \times 3^1 \times 5^2 \]

### Step 2: Prime Factorization of 210
Now, we will find the prime factors of 210.
- Start with the smallest prime number, which is 2:
- 210 ÷ 2 = 105
- Next, divide 105 by the next smallest prime number, which is 3:
- 105 ÷ 3 = 35
- Now, divide 35 by the next smallest prime number, which is 5:
- 35 ÷ 5 = 7
- Finally, divide 7 by itself:
- 7 ÷ 7 = 1

So, the prime factorization of 210 is:
\[ 210 = 2^1 \times 3^1 \times 5^1 \times 7^1 \]

### Step 3: Identify the Maximum Power of Each Prime Factor
Next, we will identify the maximum power of each prime factor from both numbers:
- For the prime factor 2: max power is \( 2^1 \) (from both 150 and 210)
- For the prime factor 3: max power is \( 3^1 \) (from both 150 and 210)
- For the prime factor 5: max power is \( 5^2 \) (from 150)
- For the prime factor 7: max power is \( 7^1 \) (from 210)

### Step 4: Calculate the LCM
Now, we multiply these maximum powers together to find the LCM:
\[ \text{LCM} = 2^1 \times 3^1 \times 5^2 \times 7^1 \]

Calculating this step-by-step:
- \( 2^1 = 2 \)
- \( 3^1 = 3 \)
- \( 5^2 = 25 \)
- \( 7^1 = 7 \)

Now, multiply them together:
\[ \text{LCM} = 2 \times 3 \times 25 \times 7 \]

Calculating it step-by-step:
1. \( 2 \times 3 = 6 \)
2. \( 6 \times 25 = 150 \)
3. \( 150 \times 7 = 1050 \)

Thus, the LCM of 150 and 210 is:
\[ \text{LCM} = 1050 \]

### Final Answer
The LCM of 150 and 210 is **1050**.

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