The LCM of 72 and 84 is :

Options

A. `12 xx 6 xx 7`
B. `6 xx 2 xx 7`
C. `12 xx 7 xx 3`
D. `72 xx 84`

Answer

To find the LCM (Least Common Multiple) of 72 and 84, we will follow these steps:

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

**For 72:**
- Divide by 2: \( 72 \div 2 = 36 \)
- Divide by 2: \( 36 \div 2 = 18 \)
- Divide by 2: \( 18 \div 2 = 9 \)
- Divide by 3: \( 9 \div 3 = 3 \)
- Divide by 3: \( 3 \div 3 = 1 \)

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

**For 84:**
- Divide by 2: \( 84 \div 2 = 42 \)
- Divide by 2: \( 42 \div 2 = 21 \)
- Divide by 3: \( 21 \div 3 = 7 \)
- Divide by 7: \( 7 \div 7 = 1 \)

So, the prime factorization of 84 is:
\[ 84 = 2^2 \times 3^1 \times 7^1 \]

### Step 2: Identify the Highest Powers
Next, we will identify the highest powers of each prime factor from both factorizations:
- For \(2\): The highest power is \(2^3\) (from 72).
- For \(3\): The highest power is \(3^2\) (from 72).
- For \(7\): The highest power is \(7^1\) (from 84).

### Step 3: Calculate the LCM
Now, we can calculate the LCM by multiplying these highest powers together:
\[ LCM = 2^3 \times 3^2 \times 7^1 \]

Calculating this step-by-step:
- \(2^3 = 8\)
- \(3^2 = 9\)
- \(7^1 = 7\)

Now multiply these results:
\[ LCM = 8 \times 9 \times 7 \]

Calculating \(8 \times 9 = 72\), then:
\[ 72 \times 7 = 504 \]

Thus, the LCM of 72 and 84 is:
\[ \text{LCM} = 504 \]

### Final Answer
The LCM of 72 and 84 is **504**.

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