Find HCF and LCM of 56 and 112 by prime factorization method.

Answer

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of 56 and 112 using the prime factorization method, follow these steps:

### Step 1: Prime Factorization of 56
1. Start dividing 56 by the smallest prime number, which is 2.
- \( 56 \div 2 = 28 \)
2. Divide 28 by 2 again.
- \( 28 \div 2 = 14 \)
3. Divide 14 by 2 once more.
- \( 14 \div 2 = 7 \)
4. Finally, 7 is a prime number, so we stop here.

Thus, the prime factorization of 56 is:
\[ 56 = 2^3 \times 7^1 \]

### Step 2: Prime Factorization of 112
1. Start dividing 112 by 2.
- \( 112 \div 2 = 56 \)
2. Divide 56 by 2 again.
- \( 56 \div 2 = 28 \)
3. Divide 28 by 2.
- \( 28 \div 2 = 14 \)
4. Divide 14 by 2.
- \( 14 \div 2 = 7 \)
5. Finally, 7 is a prime number, so we stop here.

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

### Step 3: Finding HCF
To find the HCF, we take the lowest power of each common prime factor from the factorizations:
- For the prime factor 2: the minimum power is \( 2^3 \)
- For the prime factor 7: the minimum power is \( 7^1 \)

So, the HCF is:
\[ \text{HCF} = 2^3 \times 7^1 = 8 \times 7 = 56 \]

### Step 4: Finding LCM
To find the LCM, we take the highest power of each prime factor from the factorizations:
- For the prime factor 2: the maximum power is \( 2^4 \)
- For the prime factor 7: the maximum power is \( 7^1 \)

So, the LCM is:
\[ \text{LCM} = 2^4 \times 7^1 = 16 \times 7 = 112 \]

### Final Answer
- HCF of 56 and 112 is **56**.
- LCM of 56 and 112 is **112**.