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Answer :
To determine the Highest Common Factor (HCF) of 330 and 396 using prime factorization, follow these steps:
1. Prime Factorization of Each Number:
- 330:
- Start by dividing 330 by the smallest prime number, 2.
- 330 ÷ 2 = 165, so 2 is a factor.
- Next, check 165. It's not divisible by 2, so try the next prime number, 3.
- 165 ÷ 3 = 55, so 3 is a factor.
- Now, for 55, it's not divisible by 3, but it is by 5.
- 55 ÷ 5 = 11, so 5 is a factor.
- Lastly, 11 is a prime number itself, so it cannot be factored further.
- Prime factorization of 330 is: [tex]\(2^1 \times 3^1 \times 5^1 \times 11^1\)[/tex].
- 396:
- Begin with the smallest prime number, 2.
- 396 ÷ 2 = 198, so 2 is a factor.
- 198 ÷ 2 = 99, so another 2 is a factor (since we kept dividing by 2).
- Next, 99 is not divisible by 2, so check with 3.
- 99 ÷ 3 = 33, so 3 is a factor.
- Then, divide 33 by 3 again, which results in 11.
- 11 is a prime number itself and cannot be divided further.
- Prime factorization of 396 is: [tex]\(2^2 \times 3^2 \times 11^1\)[/tex].
2. Identify Common Factors:
- Compare the prime factorizations of both numbers:
- 330: [tex]\(2^1 \times 3^1 \times 5^1 \times 11^1\)[/tex]
- 396: [tex]\(2^2 \times 3^2 \times 11^1\)[/tex]
- The common prime factors are 2, 3, and 11.
3. Calculate the HCF:
- For each common prime factor, choose the smallest power present in both factorizations:
- For 2: the lowest power is [tex]\(2^1\)[/tex]
- For 3: the lowest power is [tex]\(3^1\)[/tex]
- For 11: the lowest power is [tex]\(11^1\)[/tex]
- Multiply these together to find the HCF:
- HCF = [tex]\(2^1 \times 3^1 \times 11^1 = 2 \times 3 \times 11 = 66\)[/tex]
Therefore, the Highest Common Factor of 330 and 396 is 66.
1. Prime Factorization of Each Number:
- 330:
- Start by dividing 330 by the smallest prime number, 2.
- 330 ÷ 2 = 165, so 2 is a factor.
- Next, check 165. It's not divisible by 2, so try the next prime number, 3.
- 165 ÷ 3 = 55, so 3 is a factor.
- Now, for 55, it's not divisible by 3, but it is by 5.
- 55 ÷ 5 = 11, so 5 is a factor.
- Lastly, 11 is a prime number itself, so it cannot be factored further.
- Prime factorization of 330 is: [tex]\(2^1 \times 3^1 \times 5^1 \times 11^1\)[/tex].
- 396:
- Begin with the smallest prime number, 2.
- 396 ÷ 2 = 198, so 2 is a factor.
- 198 ÷ 2 = 99, so another 2 is a factor (since we kept dividing by 2).
- Next, 99 is not divisible by 2, so check with 3.
- 99 ÷ 3 = 33, so 3 is a factor.
- Then, divide 33 by 3 again, which results in 11.
- 11 is a prime number itself and cannot be divided further.
- Prime factorization of 396 is: [tex]\(2^2 \times 3^2 \times 11^1\)[/tex].
2. Identify Common Factors:
- Compare the prime factorizations of both numbers:
- 330: [tex]\(2^1 \times 3^1 \times 5^1 \times 11^1\)[/tex]
- 396: [tex]\(2^2 \times 3^2 \times 11^1\)[/tex]
- The common prime factors are 2, 3, and 11.
3. Calculate the HCF:
- For each common prime factor, choose the smallest power present in both factorizations:
- For 2: the lowest power is [tex]\(2^1\)[/tex]
- For 3: the lowest power is [tex]\(3^1\)[/tex]
- For 11: the lowest power is [tex]\(11^1\)[/tex]
- Multiply these together to find the HCF:
- HCF = [tex]\(2^1 \times 3^1 \times 11^1 = 2 \times 3 \times 11 = 66\)[/tex]
Therefore, the Highest Common Factor of 330 and 396 is 66.
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