Find the LCM of the numbers given below :
60, 75

Answer

To find the Least Common Multiple (LCM) of the numbers 60 and 75, we can follow these steps:

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

**For 60:**
- Divide 60 by 2:
\( 60 \div 2 = 30 \)
- Divide 30 by 2:
\( 30 \div 2 = 15 \)
- Divide 15 by 3:
\( 15 \div 3 = 5 \)
- Divide 5 by 5:
\( 5 \div 5 = 1 \)

So, the prime factorization of 60 is:
\( 60 = 2^2 \times 3^1 \times 5^1 \)

**For 75:**
- Divide 75 by 3:
\( 75 \div 3 = 25 \)
- Divide 25 by 5:
\( 25 \div 5 = 5 \)
- Divide 5 by 5:
\( 5 \div 5 = 1 \)

So, the prime factorization of 75 is:
\( 75 = 3^1 \times 5^2 \)

### Step 2: Identify the Highest Powers
Next, we identify the highest powers of all prime factors from both factorizations.

- For 2: The highest power is \( 2^2 \) (from 60).
- For 3: The highest power is \( 3^1 \) (common in both).
- For 5: The highest power is \( 5^2 \) (from 75).

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

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

Calculating this step-by-step:
- \( 2^2 = 4 \)
- \( 3^1 = 3 \)
- \( 5^2 = 25 \)

Now, multiply them together:
\[ 4 \times 3 = 12 \]
\[ 12 \times 25 = 300 \]

Thus, the LCM of 60 and 75 is **300**.

### Summary
The LCM of 60 and 75 is **300**.

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