Find the LCM of 112 , 168 , 266 by the prime factorization method.

Answer

To find the LCM of 112, 168, and 266 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of Each Number

**For 112:**
1. Divide by 2: \( 112 \div 2 = 56 \)
2. Divide by 2: \( 56 \div 2 = 28 \)
3. Divide by 2: \( 28 \div 2 = 14 \)
4. Divide by 2: \( 14 \div 2 = 7 \)
5. 7 is a prime number.

So, the prime factorization of 112 is:
\[ 112 = 2^4 \times 7^1 \]

**For 168:**
1. Divide by 2: \( 168 \div 2 = 84 \)
2. Divide by 2: \( 84 \div 2 = 42 \)
3. Divide by 2: \( 42 \div 2 = 21 \)
4. Divide by 3: \( 21 \div 3 = 7 \)
5. 7 is a prime number.

So, the prime factorization of 168 is:
\[ 168 = 2^3 \times 3^1 \times 7^1 \]

**For 266:**
1. Divide by 2: \( 266 \div 2 = 133 \)
2. 133 is not divisible by 2, try 3 (not divisible), then 7 (not divisible), then 19: \( 133 \div 19 = 7 \)
3. 7 is a prime number.

So, the prime factorization of 266 is:
\[ 266 = 2^1 \times 7^1 \times 19^1 \]

### Step 2: Identify the Highest Power of Each Prime Factor

Now we will take the highest power of each prime factor from the factorizations:

- For \( 2 \): The highest power is \( 2^4 \) (from 112).
- For \( 3 \): The highest power is \( 3^1 \) (from 168).
- For \( 7 \): The highest power is \( 7^1 \) (common in all).
- For \( 19 \): The highest power is \( 19^1 \) (from 266).

### Step 3: Calculate the LCM

Now we can calculate the LCM by multiplying these highest powers together:

\[ \text{LCM} = 2^4 \times 3^1 \times 7^1 \times 19^1 \]

Calculating this step by step:

1. \( 2^4 = 16 \)
2. \( 16 \times 3 = 48 \)
3. \( 48 \times 7 = 336 \)
4. \( 336 \times 19 = 6384 \)

Thus, the LCM of 112, 168, and 266 is:
\[ \text{LCM} = 6384 \]

### Final Answer
The LCM of 112, 168, and 266 is **6384**.

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