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Answer :
To determine the Highest Common Factor (HCF) of 330 and 396 using prime factorization, we'll follow these steps:
1. Prime Factorize 330:
- First, divide 330 by the smallest prime number, which is 2. Since 330 is divisible by 2, we get 330 ÷ 2 = 165.
- Next, 165 is not divisible by 2, so we try the next prime number, 3. Dividing 165 by 3 gives us 165 ÷ 3 = 55.
- The number 55 is not divisible by 3, so we try the next prime number, 5. Dividing 55 by 5 gives us 55 ÷ 5 = 11.
- The number 11 is a prime itself, so the factorization stops here.
- Therefore, the prime factorization of 330 is [tex]\(2^1 \times 3^1 \times 5^1 \times 11^1\)[/tex].
2. Prime Factorize 396:
- Start by dividing 396 by 2, the smallest prime number. Since 396 is divisible by 2, we get 396 ÷ 2 = 198.
- Divide 198 by 2 again, which gives 198 ÷ 2 = 99.
- Next, 99 is not divisible by 2, so try dividing it by 3. Dividing 99 by 3 gives us 99 ÷ 3 = 33.
- 33 is divisible by 3 again, so divide by 3 to get 33 ÷ 3 = 11.
- The number 11 is a prime number, so we stop here.
- Therefore, the prime factorization of 396 is [tex]\(2^2 \times 3^2 \times 11^1\)[/tex].
3. Identify Common Factors:
- From the prime factorizations:
- 330: [tex]\(2^1, 3^1, 5^1, 11^1\)[/tex]
- 396: [tex]\(2^2, 3^2, 11^1\)[/tex]
- The common prime factors are 2, 3, and 11.
- We take the lowest power of each common factor:
- For 2, the lowest power is 1.
- For 3, the lowest power is 1.
- For 11, the lowest power is 1.
4. Calculate the HCF:
- Multiply the lowest powers of all common prime factors:
[tex]\[
\text{HCF} = 2^1 \times 3^1 \times 11^1 = 2 \times 3 \times 11 = 66
\][/tex]
Thus, the Highest Common Factor (HCF) of 330 and 396 is 66.
1. Prime Factorize 330:
- First, divide 330 by the smallest prime number, which is 2. Since 330 is divisible by 2, we get 330 ÷ 2 = 165.
- Next, 165 is not divisible by 2, so we try the next prime number, 3. Dividing 165 by 3 gives us 165 ÷ 3 = 55.
- The number 55 is not divisible by 3, so we try the next prime number, 5. Dividing 55 by 5 gives us 55 ÷ 5 = 11.
- The number 11 is a prime itself, so the factorization stops here.
- Therefore, the prime factorization of 330 is [tex]\(2^1 \times 3^1 \times 5^1 \times 11^1\)[/tex].
2. Prime Factorize 396:
- Start by dividing 396 by 2, the smallest prime number. Since 396 is divisible by 2, we get 396 ÷ 2 = 198.
- Divide 198 by 2 again, which gives 198 ÷ 2 = 99.
- Next, 99 is not divisible by 2, so try dividing it by 3. Dividing 99 by 3 gives us 99 ÷ 3 = 33.
- 33 is divisible by 3 again, so divide by 3 to get 33 ÷ 3 = 11.
- The number 11 is a prime number, so we stop here.
- Therefore, the prime factorization of 396 is [tex]\(2^2 \times 3^2 \times 11^1\)[/tex].
3. Identify Common Factors:
- From the prime factorizations:
- 330: [tex]\(2^1, 3^1, 5^1, 11^1\)[/tex]
- 396: [tex]\(2^2, 3^2, 11^1\)[/tex]
- The common prime factors are 2, 3, and 11.
- We take the lowest power of each common factor:
- For 2, the lowest power is 1.
- For 3, the lowest power is 1.
- For 11, the lowest power is 1.
4. Calculate the HCF:
- Multiply the lowest powers of all common prime factors:
[tex]\[
\text{HCF} = 2^1 \times 3^1 \times 11^1 = 2 \times 3 \times 11 = 66
\][/tex]
Thus, the Highest Common Factor (HCF) of 330 and 396 is 66.
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