Find the LCM of the numbers given below :
28, 36, 45, 60

Answer

To find the LCM (Least Common Multiple) of the numbers 28, 36, 45, and 60, we can follow these steps:

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

- **28**:
- Divide by 2: \(28 \div 2 = 14\)
- Divide by 2: \(14 \div 2 = 7\)
- 7 is a prime number.
- So, the prime factorization of 28 is \(2^2 \times 7^1\).

- **36**:
- Divide by 2: \(36 \div 2 = 18\)
- Divide by 2: \(18 \div 2 = 9\)
- Divide by 3: \(9 \div 3 = 3\)
- Divide by 3: \(3 \div 3 = 1\)
- So, the prime factorization of 36 is \(2^2 \times 3^2\).

- **45**:
- Divide by 3: \(45 \div 3 = 15\)
- Divide by 3: \(15 \div 3 = 5\)
- 5 is a prime number.
- So, the prime factorization of 45 is \(3^2 \times 5^1\).

- **60**:
- Divide by 2: \(60 \div 2 = 30\)
- Divide by 2: \(30 \div 2 = 15\)
- Divide by 3: \(15 \div 3 = 5\)
- 5 is a prime number.
- So, the prime factorization of 60 is \(2^2 \times 3^1 \times 5^1\).

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

- For \(2\): The highest power is \(2^2\) (from 28, 36, and 60).
- For \(3\): The highest power is \(3^2\) (from 36 and 45).
- For \(5\): The highest power is \(5^1\) (from 45 and 60).
- For \(7\): The highest power is \(7^1\) (from 28).

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

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

Calculating this step-by-step:

1. \(2^2 = 4\)
2. \(3^2 = 9\)
3. \(5^1 = 5\)
4. \(7^1 = 7\)

Now multiply them together:

\[ 4 \times 9 = 36 \]
\[ 36 \times 5 = 180 \]
\[ 180 \times 7 = 1260 \]

Thus, the LCM of 28, 36, 45, and 60 is **1260**.

### Summary
The LCM of the numbers 28, 36, 45, and 60 is **1260**.

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