The L.C.M. of 30 and 45 is

Options

A. 15
B. 30
C. 45
D. 90

Answer

To find the L.C.M. (Least Common Multiple) of 30 and 45, we can follow these steps:

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

- **For 30**:
- 30 can be divided by 2 (the smallest prime number):
- \( 30 \div 2 = 15 \)
- Next, we factor 15:
- 15 can be divided by 3:
- \( 15 \div 3 = 5 \)
- Now we have:
- \( 30 = 2 \times 3 \times 5 \)

- **For 45**:
- 45 can be divided by 3:
- \( 45 \div 3 = 15 \)
- Next, we factor 15 (as done before):
- \( 15 \div 3 = 5 \)
- Now we have:
- \( 45 = 3 \times 3 \times 5 \) or \( 45 = 3^2 \times 5 \)

### Step 2: Identify Common and Unique Factors
Now we will list the prime factors we found:
- Prime factors of 30: \( 2, 3, 5 \)
- Prime factors of 45: \( 3^2, 5 \)

To find the L.C.M., we take the highest power of each prime factor present in either number:
- For 2: The highest power is \( 2^1 \) (from 30)
- For 3: The highest power is \( 3^2 \) (from 45)
- For 5: The highest power is \( 5^1 \) (common in both)

### Step 3: Calculate the L.C.M.
Now we multiply these together to find the L.C.M.:
\[ \text{L.C.M.} = 2^1 \times 3^2 \times 5^1 \]
Calculating this step-by-step:
- First, calculate \( 3^2 = 9 \)
- Then, multiply by 2:
- \( 2 \times 9 = 18 \)
- Finally, multiply by 5:
- \( 18 \times 5 = 90 \)

Thus, the L.C.M. of 30 and 45 is **90**.

### Final Answer
The L.C.M. of 30 and 45 is **90**.

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