Using prime factorisation method, find the LCM or 21, 28, 36, 45.

Answer

To find the LCM of the numbers 21, 28, 36, and 45 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of Each Number
We need to find the prime factors of each number.

- **21**:
- 21 can be divided by 3 (21 ÷ 3 = 7).
- Both 3 and 7 are prime numbers.
- Therefore, the prime factorization of 21 is:
\[ 21 = 3^1 \times 7^1 \]

- **28**:
- 28 can be divided by 2 (28 ÷ 2 = 14), and then 14 can be divided by 2 again (14 ÷ 2 = 7).
- Both 2 and 7 are prime numbers.
- Therefore, the prime factorization of 28 is:
\[ 28 = 2^2 \times 7^1 \]

- **36**:
- 36 can be divided by 2 (36 ÷ 2 = 18), then 18 can be divided by 2 again (18 ÷ 2 = 9), and finally, 9 can be divided by 3 (9 ÷ 3 = 3).
- Therefore, the prime factorization of 36 is:
\[ 36 = 2^2 \times 3^2 \]

- **45**:
- 45 can be divided by 3 (45 ÷ 3 = 15), and then 15 can be divided by 3 again (15 ÷ 3 = 5).
- Therefore, the prime factorization of 45 is:
\[ 45 = 3^2 \times 5^1 \]

### Step 2: Identify the Maximum Power of Each Prime Factor
Next, we will identify the maximum power of each prime factor from all the factorizations.

- **For 2**:
- Appears in 28 and 36.
- Maximum power is \(2^2\).

- **For 3**:
- Appears in 21, 36, and 45.
- Maximum power is \(3^2\).

- **For 5**:
- Appears only in 45.
- Maximum power is \(5^1\).

- **For 7**:
- Appears in 21 and 28.
- Maximum power is \(7^1\).

### Step 3: Calculate the LCM
Now, we will multiply the prime factors raised to their maximum powers to find the LCM.

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

Calculating this step-by-step:

1. Calculate \(2^2\):
\[ 2^2 = 4 \]

2. Calculate \(3^2\):
\[ 3^2 = 9 \]

3. Multiply \(4\) and \(9\):
\[ 4 \times 9 = 36 \]

4. Multiply \(36\) by \(5\):
\[ 36 \times 5 = 180 \]

5. Finally, multiply \(180\) by \(7\):
\[ 180 \times 7 = 1260 \]

Thus, the LCM of 21, 28, 36, and 45 is:
\[ \text{LCM} = 1260 \]