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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:
- Break down each number into its prime factors.
For 330:
- Divide 330 by 2 (the smallest prime number): [tex]\(330 \div 2 = 165\)[/tex].
- Divide 165 by 3 (the next smallest prime number): [tex]\(165 \div 3 = 55\)[/tex].
- Divide 55 by 5: [tex]\(55 \div 5 = 11\)[/tex].
- Finally, 11 is a prime number.
So, the prime factorization of 330 is [tex]\(2^1 \times 3^1 \times 5^1 \times 11^1\)[/tex].
For 396:
- Divide 396 by 2: [tex]\(396 \div 2 = 198\)[/tex].
- Divide 198 by 2 again: [tex]\(198 \div 2 = 99\)[/tex].
- Divide 99 by 3: [tex]\(99 \div 3 = 33\)[/tex].
- Divide 33 by 3 again: [tex]\(33 \div 3 = 11\)[/tex].
- The last factor is 11, which is prime.
So, the prime factorization of 396 is [tex]\(2^2 \times 3^2 \times 11^1\)[/tex].
2. Identify Common Prime Factors:
- Find the common prime factors from the factorizations of both numbers.
Common factors are:
- The common prime factors between 330 and 396 are 2, 3, and 11.
3. Determine the Lowest Powers of the Common Prime Factors:
- For each of the common prime factors, take the lowest power.
- [tex]\(2^1\)[/tex] (since the minimum power of 2 is 1),
- [tex]\(3^1\)[/tex] (since the minimum power of 3 is 1),
- [tex]\(11^1\)[/tex] (since the minimum power of 11 is 1).
4. Calculate the HCF:
- Multiply these together to get the HCF.
[tex]\(HCF = 2^1 \times 3^1 \times 11^1 = 2 \times 3 \times 11 = 66\)[/tex].
Therefore, the HCF of 330 and 396 is 66.
1. Prime Factorization of Each Number:
- Break down each number into its prime factors.
For 330:
- Divide 330 by 2 (the smallest prime number): [tex]\(330 \div 2 = 165\)[/tex].
- Divide 165 by 3 (the next smallest prime number): [tex]\(165 \div 3 = 55\)[/tex].
- Divide 55 by 5: [tex]\(55 \div 5 = 11\)[/tex].
- Finally, 11 is a prime number.
So, the prime factorization of 330 is [tex]\(2^1 \times 3^1 \times 5^1 \times 11^1\)[/tex].
For 396:
- Divide 396 by 2: [tex]\(396 \div 2 = 198\)[/tex].
- Divide 198 by 2 again: [tex]\(198 \div 2 = 99\)[/tex].
- Divide 99 by 3: [tex]\(99 \div 3 = 33\)[/tex].
- Divide 33 by 3 again: [tex]\(33 \div 3 = 11\)[/tex].
- The last factor is 11, which is prime.
So, the prime factorization of 396 is [tex]\(2^2 \times 3^2 \times 11^1\)[/tex].
2. Identify Common Prime Factors:
- Find the common prime factors from the factorizations of both numbers.
Common factors are:
- The common prime factors between 330 and 396 are 2, 3, and 11.
3. Determine the Lowest Powers of the Common Prime Factors:
- For each of the common prime factors, take the lowest power.
- [tex]\(2^1\)[/tex] (since the minimum power of 2 is 1),
- [tex]\(3^1\)[/tex] (since the minimum power of 3 is 1),
- [tex]\(11^1\)[/tex] (since the minimum power of 11 is 1).
4. Calculate the HCF:
- Multiply these together to get the HCF.
[tex]\(HCF = 2^1 \times 3^1 \times 11^1 = 2 \times 3 \times 11 = 66\)[/tex].
Therefore, the HCF of 330 and 396 is 66.
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