what is the LCM of 120 and 450 ?

Options

A. 2400
B. 1800
C. 3600
D. 4800

Answer

To find the LCM (Least Common Multiple) of 120 and 450, we can follow these steps:

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

**For 120:**
- Divide by 2: \(120 \div 2 = 60\)
- Divide by 2: \(60 \div 2 = 30\)
- Divide by 2: \(30 \div 2 = 15\)
- Divide by 3: \(15 \div 3 = 5\)
- Divide by 5: \(5 \div 5 = 1\)

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

**For 450:**
- Divide by 2: \(450 \div 2 = 225\)
- Divide by 3: \(225 \div 3 = 75\)
- Divide by 3: \(75 \div 3 = 25\)
- Divide by 5: \(25 \div 5 = 5\)
- Divide by 5: \(5 \div 5 = 1\)

So, the prime factorization of 450 is:
\[ 450 = 2^1 \times 3^2 \times 5^2 \]

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

- For \(2\): The highest power is \(2^3\) (from 120).
- For \(3\): The highest power is \(3^2\) (from 450).
- For \(5\): The highest power is \(5^2\) (from 450).

### Step 3: Calculate the LCM
Now, we can calculate the LCM by multiplying these highest powers together:

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

Calculating this step-by-step:

1. Calculate \(2^3 = 8\)
2. Calculate \(3^2 = 9\)
3. Calculate \(5^2 = 25\)

Now multiply these results together:

\[ \text{LCM} = 8 \times 9 \times 25 \]

Calculating \(8 \times 9 = 72\)

Then, \(72 \times 25 = 1800\)

Thus, the LCM of 120 and 450 is:
\[ \text{LCM} = 1800 \]

### Final Answer
The LCM of 120 and 450 is **1800**.

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