Find the HCF by prime factorization. 165 and 275 .
The Correct Answer and Explanation is:
To find the Highest Common Factor (HCF) of 165 and 275 using prime factorization, begin by breaking each number into its prime factors:
Step 1: Prime factorization of 165
165 ÷ 5 = 33
33 ÷ 3 = 11
11 is a prime number.
So, the prime factorization of 165 is:
165 = 5 × 3 × 11
Step 2: Prime factorization of 275
275 ÷ 5 = 55
55 ÷ 5 = 11
11 is a prime number.
So, the prime factorization of 275 is:
275 = 5 × 5 × 11
Step 3: Identify common prime factors
Now compare the prime factors of both numbers.
- 165 has: 5, 3, 11
- 275 has: 5, 5, 11
The common prime factors are:
5 and 11
The product of these common factors is:
5 × 11 = 55
Final Answer:
HCF of 165 and 275 is 55
Explanation (300 words)
The Highest Common Factor (HCF), also known as the greatest common divisor (GCD), is the largest number that can divide two or more numbers without leaving a remainder. One reliable method to find the HCF is by using prime factorization, where each number is broken down into a product of its prime numbers.
To begin, consider the number 165. Divide it by the smallest prime number that goes into it, which is 5. Dividing 165 by 5 gives 33. Next, divide 33 by the smallest prime number that fits, which is 3, and the result is 11. Since 11 is already a prime number, the prime factorization of 165 is complete as 5 × 3 × 11.
Now move to 275. Dividing 275 by 5 gives 55. Then dividing 55 by 5 again gives 11. Since 11 is a prime number, the prime factorization of 275 is 5 × 5 × 11.
Next, compare the two sets of prime factors. Both 165 and 275 have the prime numbers 5 and 11 in common. While 275 has two 5s, 165 has only one. The HCF includes only the prime factors that appear in both factorizations, taken to the lowest power. Therefore, take one 5 and one 11.
Multiplying these gives 5 × 11 = 55. This value is the largest number that can divide both 165 and 275 exactly, with no remainder.
Thus, the HCF of 165 and 275 is 55.
