Finding the Highest Common Factor (HCF) of 72 and 90: A full breakdown
Finding the highest common factor (HCF), also known as the greatest common divisor (GCD), of two numbers is a fundamental concept in mathematics. This guide will explore multiple methods for determining the HCF of 72 and 90, explaining the underlying principles and providing a deeper understanding of this important mathematical operation. We'll move beyond simply finding the answer and break down the 'why' behind the methods, making this concept accessible to everyone, from beginners to those looking to refresh their knowledge.
Not obvious, but once you see it — you'll see it everywhere.
Introduction: Understanding HCF
The highest common factor (HCF) of two or more numbers is the largest number that divides each of them without leaving a remainder. Practically speaking, in simpler terms, it's the biggest number that's a factor of both numbers. Consider this: for example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The common factors of 12 and 18 are 1, 2, 3, and 6. Which means, the highest common factor (HCF) of 12 and 18 is 6.
This concept is crucial in various mathematical applications, including simplification of fractions, solving algebraic equations, and understanding number theory. Let's now apply this understanding to find the HCF of 72 and 90 Worth keeping that in mind..
Method 1: Prime Factorization
This method involves breaking down each number into its prime factors. That's why prime numbers are numbers greater than 1 that are only divisible by 1 and themselves (e. Because of that, g. , 2, 3, 5, 7, 11, etc.) Still holds up..
- Prime Factorization of 72:
We can start by dividing 72 by the smallest prime number, 2:
72 ÷ 2 = 36 36 ÷ 2 = 18 18 ÷ 2 = 9 9 ÷ 3 = 3 3 ÷ 3 = 1
Because of this, the prime factorization of 72 is 2 x 2 x 2 x 3 x 3 = 2³ x 3²
- Prime Factorization of 90:
Let's do the same for 90:
90 ÷ 2 = 45 45 ÷ 3 = 15 15 ÷ 3 = 5 5 ÷ 5 = 1
So, the prime factorization of 90 is 2 x 3 x 3 x 5 = 2 x 3² x 5
- Finding the HCF:
Now, we identify the common prime factors and their lowest powers present in both factorizations:
Both 72 and 90 have a common factor of 2 (to the power of 1) and a common factor of 3 (to the power of 2) Simple, but easy to overlook. Which is the point..
That's why, the HCF of 72 and 90 is 2¹ x 3² = 2 x 9 = 18.
Method 2: Listing Factors
This method involves listing all the factors of each number and then identifying the common factors. While straightforward for smaller numbers, it can become tedious for larger ones.
- Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
- Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Comparing the two lists, the common factors are 1, 2, 3, 6, 9, and 18. Even so, the highest of these common factors is 18. So, the HCF of 72 and 90 is 18.
Method 3: Euclidean Algorithm
This is a highly efficient algorithm, especially for larger numbers. It involves repeatedly applying the division algorithm until the remainder is 0. The last non-zero remainder is the HCF.
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Divide the larger number (90) by the smaller number (72): 90 ÷ 72 = 1 with a remainder of 18
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Replace the larger number with the smaller number (72) and the smaller number with the remainder (18): 72 ÷ 18 = 4 with a remainder of 0
Since the remainder is 0, the HCF is the last non-zero remainder, which is 18.
Method 4: Using the Formula (for two numbers only)
While not as intuitive as the other methods, a formula exists for calculating the HCF of two numbers using their least common multiple (LCM). The relationship is as follows:
HCF(a, b) x LCM(a, b) = a x b
Where 'a' and 'b' are the two numbers.
First, we need to find the LCM of 72 and 90. We can use the prime factorization method:
- Prime factorization of 72: 2³ x 3²
- Prime factorization of 90: 2 x 3² x 5
The LCM is found by taking the highest power of each prime factor present in either factorization: 2³ x 3² x 5 = 360
Now, we can use the formula:
HCF(72, 90) x 360 = 72 x 90 HCF(72, 90) = (72 x 90) / 360 HCF(72, 90) = 18
This method highlights the interconnectedness between HCF and LCM That's the part that actually makes a difference..
Explanation of the Methods and their Efficiency
Each method offers a unique approach to finding the HCF. That's why the listing factors method is simple for smaller numbers but becomes impractical for larger ones. The best method depends on the context and the size of the numbers involved. The Euclidean algorithm is computationally efficient, especially for large numbers, requiring fewer steps. Day to day, the prime factorization method provides a deeper understanding of the numbers' composition. On top of that, the formula method is concise but requires prior knowledge of the LCM. For relatively small numbers like 72 and 90, the prime factorization or listing factors method might be sufficient, while for larger numbers, the Euclidean algorithm offers a more efficient solution That alone is useful..
Frequently Asked Questions (FAQ)
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Q: What is the difference between HCF and LCM?
A: The HCF is the largest number that divides both numbers without leaving a remainder, while the LCM is the smallest number that is a multiple of both numbers. They are inversely related.
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Q: Can the HCF of two numbers be 1?
A: Yes, if the two numbers are coprime (meaning they share no common factors other than 1). Here's one way to look at it: the HCF of 15 and 28 is 1 And that's really what it comes down to..
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Q: What if I want to find the HCF of more than two numbers?
A: You can extend any of the methods above. For prime factorization and the Euclidean algorithm, you simply find the HCF of the first two numbers, then find the HCF of that result and the next number, and so on. For the listing factors method, you need to find the common factors among all the numbers.
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Q: Is there a way to find the HCF using a calculator?
A: Many scientific calculators have a built-in function to calculate the GCD (greatest common divisor), which is the same as the HCF.
Conclusion: Mastering HCF Calculations
Finding the highest common factor is a fundamental skill in mathematics with broad applications. This guide has explored various methods for calculating the HCF of 72 and 90, demonstrating the underlying principles and their relative efficiency. Day to day, understanding these methods empowers you to tackle similar problems with confidence, regardless of the size of the numbers involved. Remember to choose the method that best suits your needs and the numbers you're working with. Still, practice is key to mastering this important mathematical concept. By understanding the "why" behind the calculations, you'll not only be able to find the HCF but also appreciate the elegance and logic within mathematical processes.
