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:**
- 240 is even, so we divide by 2:
- \(240 \div 2 = 120\)
- \(120 \div 2 = 60\)
- \(60 \div 2 = 30\)
- \(30 \div 2 = 15\)
- Now, 15 is not even, so we divide by the next prime number, which is 3:
- \(15 \div 3 = 5\)
- Finally, 5 is a prime number.

So, the prime factorization of 240 is:
\[ 240 = 2^4 \times 3^1 \times 5^1 \]

**For 924:**
- 924 is also even, so we divide by 2:
- \(924 \div 2 = 462\)
- \(462 \div 2 = 231\)
- Now, 231 is not even, so we divide by the next prime number, which is 3:
- \(231 \div 3 = 77\)
- Next, we divide 77 by the next prime number, which is 7:
- \(77 \div 7 = 11\)
- Finally, 11 is a prime number.

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 identify the highest powers of all prime factors present in 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.

\[ \text{LCM} = 2^4 \times 3^1 \times 5^1 \times 7^1 \times 11^1 \]

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

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

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

### Final Answer
The Lowest Common Multiple (LCM) of 240 and 924 is **18480**.

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