Hcf Of 90 And 396

Finding the Highest Common Factor (HCF) of 90 and 396: A complete walkthrough

Finding the Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two numbers is a fundamental concept in mathematics. This article provides a complete walkthrough to calculating the HCF of 90 and 396, exploring various methods and explaining the underlying mathematical principles. Understanding HCF is crucial for simplifying fractions, solving algebraic equations, and tackling more complex mathematical problems. We will cover different approaches, from prime factorization to the Euclidean algorithm, ensuring you gain a thorough understanding of this important concept.

Introduction to Highest Common Factor (HCF)

The HCF of two or more numbers is the largest number that divides each of them without leaving a remainder. In simpler terms, it's the biggest number that's a factor of both numbers. So naturally, 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. The highest of these common factors is 6, so the HCF of 12 and 18 is 6. This article will demonstrate how to find the HCF of 90 and 396 using several methods.

Method 1: Prime Factorization

Prime factorization involves expressing a number as a product of its prime factors. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself (e.g., 2, 3, 5, 7, 11...Consider this: ). This method is particularly useful for smaller numbers.

Step 1: Find the prime factorization of 90:

90 can be broken down as follows:

90 = 2 x 45 = 2 x 3 x 15 = 2 x 3 x 3 x 5 = 2¹ x 3² x 5¹

Step 2: Find the prime factorization of 396:

396 can be broken down as follows:

396 = 2 x 198 = 2 x 2 x 99 = 2 x 2 x 9 x 11 = 2 x 2 x 3 x 3 x 11 = 2² x 3² x 11¹

Step 3: Identify common prime factors:

Both 90 and 396 share the prime factors 2 and 3.

Step 4: Find the lowest power of each common prime factor:

The lowest power of 2 is 2¹ = 2. The lowest power of 3 is 3² = 9.

Step 5: Multiply the lowest powers together:

HCF(90, 396) = 2¹ x 3² = 2 x 9 = 18

Which means, the HCF of 90 and 396 is 18.

Method 2: Listing Factors

This method involves listing all the factors of each number and then identifying the highest common factor. While straightforward for smaller numbers, it becomes less efficient for larger numbers.

Step 1: List the factors of 90:

1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

Step 2: List the factors of 396:

1, 2, 3, 4, 6, 9, 11, 12, 18, 22, 33, 36, 44, 66, 99, 132, 198, 396

Step 3: Identify common factors:

The common factors of 90 and 396 are 1, 2, 3, 6, 9, 18.

Step 4: Determine the highest common factor:

The highest common factor among these is 18.

Which means, the HCF of 90 and 396 is 18.

Method 3: Euclidean Algorithm

The Euclidean algorithm is a highly efficient method for finding the HCF of two numbers, especially for larger numbers. And it's based on the principle that the HCF of two numbers doesn't change if the larger number is replaced by its difference with the smaller number. This process is repeated until the two numbers are equal, and that number is the HCF.

Step 1: Divide the larger number (396) by the smaller number (90):

396 ÷ 90 = 4 with a remainder of 36

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Step 2: Replace the larger number with the remainder:

Now we find the HCF of 90 and 36.

Step 3: Repeat the division process:

90 ÷ 36 = 2 with a remainder of 18

Step 4: Continue until the remainder is 0:

36 ÷ 18 = 2 with a remainder of 0

Step 5: The last non-zero remainder is the HCF:

The last non-zero remainder is 18.

So, the HCF of 90 and 396 is 18. The Euclidean algorithm provides a systematic and efficient way to find the HCF, regardless of the size of the numbers involved.

Understanding the Significance of HCF

The HCF has numerous applications across various mathematical areas. Some key applications include:

• Simplifying Fractions: The HCF helps simplify fractions to their lowest terms. To give you an idea, the fraction 90/396 can be simplified by dividing both the numerator and the denominator by their HCF (18), resulting in the equivalent fraction 5/22.

• Solving Algebraic Equations: HCF plays a role in solving Diophantine equations, which are equations where only integer solutions are sought.

• Number Theory: HCF is a fundamental concept in number theory, used in various theorems and proofs.

• Real-world Applications: HCF has practical applications in areas such as tiling, dividing objects into equal groups, and various measurement problems.

Frequently Asked Questions (FAQ)

Q1: What is the difference between HCF and LCM?

So, the Highest Common Factor (HCF) is the largest number that divides both numbers without leaving a remainder, while the Least Common Multiple (LCM) is the smallest number that is a multiple of both numbers. They are inversely related; for two numbers a and b, HCF(a, b) * LCM(a, b) = a * b.

Q2: Can the HCF of two numbers be 1?

Yes, if two numbers have no common factors other than 1, their HCF is 1. Such numbers are called coprime or relatively prime.

Q3: How do I find the HCF of more than two numbers?

To find the HCF of more than two numbers, you can use any of the methods discussed above, applying them sequentially. As an example, to find the HCF of 90, 396, and another number, you would first find the HCF of 90 and 396 (which is 18), and then find the HCF of 18 and the third number.

Q4: Is there a limitation to the Euclidean Algorithm?

While extremely efficient, the Euclidean Algorithm's computational complexity is logarithmic. This means the number of steps required grows slowly even with extremely large numbers, making it suitable even for computer calculations with very large inputs. That said, for extremely large numbers (beyond the capacity of standard computer arithmetic), more advanced techniques might be needed.

Conclusion

Finding the Highest Common Factor is a crucial skill in mathematics, with applications ranging from simplifying fractions to solving complex equations. This article has detailed three methods – prime factorization, listing factors, and the Euclidean algorithm – for calculating the HCF of 90 and 396, showcasing their individual strengths and weaknesses. The Euclidean algorithm emerges as the most efficient method, particularly for larger numbers. In practice, understanding these methods will equip you with the tools to confidently tackle HCF problems and appreciate its significance in various mathematical contexts. Remember to choose the method most suitable for the numbers involved, and practice regularly to solidify your understanding.