L.C.M of 72 and 120 is

Options

A. 240
B. 360
C. 120
D. 152

Answer

To find the Least Common Multiple (LCM) of 72 and 120, we can use the prime factorization method. Here’s a step-by-step solution:

### Step 1: Prime Factorization of 72
1. Divide 72 by the smallest prime number, which is 2:
- \( 72 \div 2 = 36 \)
2. Divide 36 by 2:
- \( 36 \div 2 = 18 \)
3. Divide 18 by 2:
- \( 18 \div 2 = 9 \)
4. Now, 9 cannot be divided by 2, so we move to the next prime number, which is 3:
- \( 9 \div 3 = 3 \)
5. Finally, divide 3 by 3:
- \( 3 \div 3 = 1 \)

Thus, the prime factorization of 72 is:
\[ 72 = 2^3 \times 3^2 \]

### Step 2: Prime Factorization of 120
1. Divide 120 by 2:
- \( 120 \div 2 = 60 \)
2. Divide 60 by 2:
- \( 60 \div 2 = 30 \)
3. Divide 30 by 2:
- \( 30 \div 2 = 15 \)
4. Now, 15 cannot be divided by 2, so we move to the next prime number, which is 3:
- \( 15 \div 3 = 5 \)
5. Finally, divide 5 by 5:
- \( 5 \div 5 = 1 \)

Thus, the prime factorization of 120 is:
\[ 120 = 2^3 \times 3^1 \times 5^1 \]

### Step 3: Finding the LCM
To find the LCM, we take the highest power of each prime factor from both numbers:
- For the prime number 2: the highest power is \( 2^3 \)
- For the prime number 3: the highest power is \( 3^2 \)
- For the prime number 5: the highest power is \( 5^1 \)

So, the LCM is:
\[ \text{LCM} = 2^3 \times 3^2 \times 5^1 \]

### Step 4: Calculating the LCM
1. Calculate \( 2^3 = 8 \)
2. Calculate \( 3^2 = 9 \)
3. Calculate \( 5^1 = 5 \)

Now multiply these results:
\[ \text{LCM} = 8 \times 9 \times 5 \]

Calculating step by step:
1. \( 8 \times 9 = 72 \)
2. \( 72 \times 5 = 360 \)

Thus, the LCM of 72 and 120 is:
\[ \text{LCM} = 360 \]

### Final Answer:
The LCM of 72 and 120 is **360**.

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