Find the Lowest Common Multiple (LCM) of 240 and 924.

Options

A. 16
B. 24
C. 18480
D. 20402

Answer

To find the Lowest Common Multiple (LCM) of 240 and 924, we can follow these steps:

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

**For 240:**
- Divide by 2: \(240 \div 2 = 120\)
- 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 240 is:
\[ 240 = 2^4 \times 3^1 \times 5^1 \]

**For 924:**
- Divide by 2: \(924 \div 2 = 462\)
- Divide by 2: \(462 \div 2 = 231\)
- Divide by 3: \(231 \div 3 = 77\)
- Divide by 7: \(77 \div 7 = 11\)
- Divide by 11: \(11 \div 11 = 1\)

So, the prime factorization of 924 is:
\[ 924 = 2^2 \times 3^1 \times 7^1 \times 11^1 \]

### Step 2: Identify the Highest Powers
Next, we take the highest power of each prime factor from both factorizations:
- For \(2\): The highest power is \(2^4\) (from 240).
- For \(3\): The highest power is \(3^1\) (common in both).
- For \(5\): The highest power is \(5^1\) (from 240).
- For \(7\): The highest power is \(7^1\) (from 924).
- For \(11\): The highest power is \(11^1\) (from 924).

### Step 3: Calculate the LCM
Now, we multiply these highest powers together to find the LCM:
\[ LCM = 2^4 \times 3^1 \times 5^1 \times 7^1 \times 11^1 \]

Calculating this step-by-step:
- \(2^4 = 16\)
- \(3^1 = 3\)
- \(5^1 = 5\)
- \(7^1 = 7\)
- \(11^1 = 11\)

Now multiply these values:
\[ LCM = 16 \times 3 = 48 \]
\[ 48 \times 5 = 240 \]
\[ 240 \times 7 = 1680 \]
\[ 1680 \times 11 = 18480 \]

Thus, the LCM of 240 and 924 is:
\[ \boxed{18480} \]