L.C.M. of 45 and 75 is:

Options

A. 215
B. 225
C. 205
D. 235

Answer

To find the L.C.M. (Least Common Multiple) of 45 and 75, we can follow these steps:

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

- **For 45:**
- 45 can be divided by 3:
- \( 45 \div 3 = 15 \)
- 15 can also be divided by 3:
- \( 15 \div 3 = 5 \)
- 5 is a prime number.

Therefore, the prime factorization of 45 is:
\[ 45 = 3^2 \times 5^1 \]

- **For 75:**
- 75 can be divided by 3:
- \( 75 \div 3 = 25 \)
- 25 can be divided by 5:
- \( 25 \div 5 = 5 \)
- 5 is a prime number.

Therefore, the prime factorization of 75 is:
\[ 75 = 3^1 \times 5^2 \]

### Step 2: Identify the Highest Powers of Each Prime Factor
Next, we take the highest power of each prime factor from both factorizations.

- For the prime factor 3:
- The highest power is \( 3^2 \) (from 45).

- For the prime factor 5:
- The highest power is \( 5^2 \) (from 75).

### Step 3: Calculate the L.C.M.
Now we can calculate the L.C.M. by multiplying these highest powers together.

\[ \text{L.C.M.} = 3^2 \times 5^2 \]

Calculating this gives:
\[ 3^2 = 9 \quad \text{and} \quad 5^2 = 25 \]
\[ \text{L.C.M.} = 9 \times 25 = 225 \]

### Final Answer
Thus, the L.C.M. of 45 and 75 is:
\[ \boxed{225} \]