Find the HCF of the following numbers by prime factorization method. (a) 76, 28 (b) 24, 16, 36 (c) 38, 64, 82
The Correct Answer and Explanation is:
Let’s find the HCF (Highest Common Factor) of the given numbers using the prime factorization method.
(a) 76 and 28
Prime factorization:
- 76 = 2 × 2 × 19 = 2² × 19
- 28 = 2 × 2 × 7 = 2² × 7
Common prime factors: 2²
So, HCF = 2² = 4
(b) 24, 16, and 36
Prime factorizations:
- 24 = 2 × 2 × 2 × 3 = 2³ × 3
- 16 = 2 × 2 × 2 × 2 = 2⁴
- 36 = 2 × 2 × 3 × 3 = 2² × 3²
Common prime factor: 2 (only base 2 is common in all)
Take the lowest exponent of 2, which is 2¹
So, HCF = 2
(c) 38, 64, and 82
Prime factorizations:
- 38 = 2 × 19
- 64 = 2 × 2 × 2 × 2 × 2 × 2 = 2⁶
- 82 = 2 × 41
Common prime factor: 2 (only 2 appears in all three)
Lowest exponent of 2 is 2¹
So, HCF = 2
Explanation:
The Highest Common Factor (HCF) is the greatest number that divides all the given numbers exactly. One of the most reliable ways to find the HCF is by using prime factorization, where we express each number as a product of its prime numbers.
To start, we break down each number into its prime factors. Then, we identify which prime factors are common to all numbers in the set. After that, we select the lowest power (exponent) of each common prime factor. Finally, we multiply these lowest powers together to find the HCF.
Let us apply this step by step:
In part (a), both 76 and 28 have the prime number 2 as a common factor. Since the highest power of 2 common to both is 2², the HCF is 4.
In part (b), 24, 16, and 36 all contain the prime number 2, but with different powers. 24 has 2³, 16 has 2⁴, and 36 has 2². The smallest power is 2¹, so HCF is 2. Other prime factors such as 3 are not common to all three numbers, so we ignore them.
In part (c), although the numbers are quite different, all have 2 as a common prime factor. Even though 64 has many twos, we only take the lowest common count, which is 1, giving an HCF of 2.
This method is systematic, reduces error, and helps with large numbers
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