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Answer :
To determine the Highest Common Factor (HCF) of 330 and 396 using prime factorization, we can follow these steps:
1. Prime Factorization of Each Number:
- 330: Start by dividing 330 by the smallest prime number, 2:
- [tex]\( 330 \div 2 = 165 \)[/tex] (so, 2 is a factor)
- 165 is not divisible by 2. Next, try the next smallest prime number, 3:
- [tex]\( 165 \div 3 = 55 \)[/tex] (so, 3 is a factor)
- 55 is not divisible by 3. Try the next smallest prime number, 5:
- [tex]\( 55 \div 5 = 11 \)[/tex] (so, 5 is a factor)
- 11 is a prime number, so it is also a factor.
Therefore, the prime factors of 330 are [tex]\( 2, 3, 5, \)[/tex] and [tex]\( 11 \)[/tex].
- 396: Start by dividing 396 by the smallest prime number, 2:
- [tex]\( 396 \div 2 = 198 \)[/tex] (so, 2 is a factor)
- Divide 198 by 2 again:
- [tex]\( 198 \div 2 = 99 \)[/tex] (so, another 2 is a factor)
- 99 is not divisible by 2. Next, try the next smallest prime number, 3:
- [tex]\( 99 \div 3 = 33 \)[/tex] (so, 3 is a factor)
- Divide 33 by 3 again:
- [tex]\( 33 \div 3 = 11 \)[/tex] (so, another 3 is a factor)
- 11 is a prime number, so it is also a factor.
Therefore, the prime factors of 396 are [tex]\( 2, 2, 3, 3, \)[/tex] and [tex]\( 11 \)[/tex].
2. Identify Common Prime Factors:
The prime factors of both 330 and 396 include [tex]\( 2, 3, \)[/tex] and [tex]\( 11 \)[/tex].
3. Calculate the HCF:
To find the HCF, take the lowest power of each common prime factor:
- 2: Appears once in 330 and twice in 396. So, use [tex]\( 2^1 \)[/tex].
- 3: Appears once in 330 and twice in 396. So, use [tex]\( 3^1 \)[/tex].
- 11: Appears once in each, so use [tex]\( 11^1 \)[/tex].
Multiply these together to get the HCF:
[tex]\[
\text{HCF} = 2^1 \times 3^1 \times 11^1 = 2 \times 3 \times 11 = 66
\][/tex]
Thus, the HCF of 330 and 396 is 66.
1. Prime Factorization of Each Number:
- 330: Start by dividing 330 by the smallest prime number, 2:
- [tex]\( 330 \div 2 = 165 \)[/tex] (so, 2 is a factor)
- 165 is not divisible by 2. Next, try the next smallest prime number, 3:
- [tex]\( 165 \div 3 = 55 \)[/tex] (so, 3 is a factor)
- 55 is not divisible by 3. Try the next smallest prime number, 5:
- [tex]\( 55 \div 5 = 11 \)[/tex] (so, 5 is a factor)
- 11 is a prime number, so it is also a factor.
Therefore, the prime factors of 330 are [tex]\( 2, 3, 5, \)[/tex] and [tex]\( 11 \)[/tex].
- 396: Start by dividing 396 by the smallest prime number, 2:
- [tex]\( 396 \div 2 = 198 \)[/tex] (so, 2 is a factor)
- Divide 198 by 2 again:
- [tex]\( 198 \div 2 = 99 \)[/tex] (so, another 2 is a factor)
- 99 is not divisible by 2. Next, try the next smallest prime number, 3:
- [tex]\( 99 \div 3 = 33 \)[/tex] (so, 3 is a factor)
- Divide 33 by 3 again:
- [tex]\( 33 \div 3 = 11 \)[/tex] (so, another 3 is a factor)
- 11 is a prime number, so it is also a factor.
Therefore, the prime factors of 396 are [tex]\( 2, 2, 3, 3, \)[/tex] and [tex]\( 11 \)[/tex].
2. Identify Common Prime Factors:
The prime factors of both 330 and 396 include [tex]\( 2, 3, \)[/tex] and [tex]\( 11 \)[/tex].
3. Calculate the HCF:
To find the HCF, take the lowest power of each common prime factor:
- 2: Appears once in 330 and twice in 396. So, use [tex]\( 2^1 \)[/tex].
- 3: Appears once in 330 and twice in 396. So, use [tex]\( 3^1 \)[/tex].
- 11: Appears once in each, so use [tex]\( 11^1 \)[/tex].
Multiply these together to get the HCF:
[tex]\[
\text{HCF} = 2^1 \times 3^1 \times 11^1 = 2 \times 3 \times 11 = 66
\][/tex]
Thus, the HCF of 330 and 396 is 66.
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