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Answer :
To find the Highest Common Factor (HCF) of 195 and 330, also known as the greatest common divisor (GCD), we can employ either the prime factorization method or the Euclidean algorithm. Here, I'll demonstrate both methods.
Method 1: Prime Factorization
Find the prime factors of each number.
195 can be factorized as follows:
[tex]195 \div 3 = 65[/tex]
65 is further factorized as:
[tex]65 \div 5 = 13[/tex]
Thus, 195 can be written as:
[tex]195 = 3 \times 5 \times 13[/tex]330 can be factorized as follows:
[tex]330 \div 2 = 165[/tex]
165 is further factorized as:
[tex]165 \div 3 = 55[/tex]
55 is further factorized as:
[tex]55 \div 5 = 11[/tex]
Thus, 330 can be written as:
[tex]330 = 2 \times 3 \times 5 \times 11[/tex]
Identify the common prime factors.
- The common prime factors of 195 and 330 are 3 and 5.
Multiply the common factors.
- [tex]\text{HCF} = 3 \times 5 = 15[/tex]
Method 2: Euclidean Algorithm
Apply the Euclidean algorithm by using division.
First, divide 330 by 195 and find the remainder:
[tex]330 \div 195 = 1 \quad \text{remainder:} \; 330 - 195 \times 1 = 135[/tex]Next, divide 195 by the remainder 135:
[tex]195 \div 135 = 1 \quad \text{remainder:} \; 195 - 135 \times 1 = 60[/tex]Then, divide 135 by 60:
[tex]135 \div 60 = 2 \quad \text{remainder:} \; 135 - 60 \times 2 = 15[/tex]Finally, divide 60 by 15:
[tex]60 \div 15 = 4 \quad \text{remainder:} \; 60 - 15 \times 4 = 0[/tex]
Conclude that the last non-zero remainder is the HCF.
- Thus, the [tex]\text{HCF}[/tex] of 195 and 330 is 15.
Both methods give us the same result: the Highest Common Factor of 195 and 330 is 15. This means that 15 is the largest number that can divide both 195 and 330 without leaving a remainder. The HCF is useful in simplifying fractions and solving problems that require finding common dimensions, lengths, or periods.
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