To find the Highest Common Factor (HCF) of two numbers using the prime factorization method, we need to follow these steps:

1. Perform Prime Factorization of each number.
2. Identify Common Prime Factors.
3. Multiply the Common Prime Factors to get the HCF.

Let's apply this method to find the HCF of 612 and 816.

Step 1: Prime Factorization

Prime Factorization of 612:

1. Divide by 2 (smallest prime number): $612 \div 2 = 306$

2. Continue dividing by 2: $306 \div 2 = 153$

3. 153 is odd, so divide by 3: $153 \div 3 = 51$

4. Continue dividing by 3: $51 \div 3 = 17$

5. 17 is a prime number.

So, the prime factorization of 612 is: $612 = 2^2 \times 3^2 \times 17$

Prime Factorization of 816:

1. Divide by 2: $816 \div 2 = 408$

2. Continue dividing by 2: $408 \div 2 = 204$

3. Continue dividing by 2: $204 \div 2 = 102$

4. Continue dividing by 2: $102 \div 2 = 51$

5. 51 is odd, so divide by 3: $51 \div 3 = 17$

6. 17 is a prime number.

So, the prime factorization of 816 is: $816 = 2^4 \times 3 \times 17$

Step 2: Identify Common Prime Factors

From the factorizations:

• 612 = $2^2 \times 3^2 \times 17$
• 816 = $2^4 \times 3 \times 17$

The common prime factors are 2, 3, and 17.

Step 3: Multiply the Common Prime Factors

Take the lowest power of each common prime factor:

• For 2, the lowest power is $2^2$.
• For 3, the lowest power is $3^1$.
• For 17, the lowest power is $17^1$.

Therefore, the HCF is:

$HCF = 2^2 \times 3 \times 17$

Let's calculate this:

$2^2 = 4$ $4 \times 3 = 12$ $12 \times 17 = 204$

Conclusion

The HCF of 612 and 816 using the prime factorization method is 204.

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Would you like any further details or have any questions about this process?

Relative Questions

1. How do you perform prime factorization for larger numbers?
2. What are other methods to find the HCF besides prime factorization?
3. How is the HCF useful in solving real-world problems?
4. Can the HCF of two numbers be larger than either of the numbers?
5. What is the relationship between HCF and LCM?
6. How can prime factorization be used to find the Least Common Multiple (LCM)?
7. What are some tips for identifying prime numbers quickly?
8. Why is it important to understand the concept of prime factorization?

Tip

When dealing with prime factorization, always start dividing by the smallest prime number (2) and work your way up to ensure you don’t miss any factors.