The Power of Common Factors: Unveiling the HCF of 777 and 1147

Introduction
The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is a fundamental concept in number theory. It represents the largest positive integer that divides two or more numbers without leaving a remainder. Understanding the HCF is crucial for solving a wide range of mathematical problems, from simplifying fractions to finding the shortest repeating cycle in a decimal expansion.

HCF of 777 and 1147
The HCF of 777 and 1147 is the largest positive integer that divides both numbers without leaving a remainder. Using prime factorization, we can find the HCF as follows:

777 = 3^2 * 7^2
1147 = 3^3 * 11

Therefore, the HCF of 777 and 1147 is 3^2 = 9.

Applications of HCF
The HCF has numerous applications in various fields, including:

• Simplifying Fractions: The HCF is used to reduce fractions to their simplest form. For example, to simplify the fraction 18/27, we find the HCF of 18 and 27, which is 9. Dividing both the numerator and denominator by 9 simplifies the fraction to 2/3.
• Finding the Shortest Repeating Cycle of a Decimal Expansion: The HCF is used to determine the length of the shortest repeating cycle in a decimal expansion. For example, the decimal expansion of 1/7 can be written as 0.142857142857..., where the sequence of digits 142857 repeats indefinitely. The HCF of 7 and 99 (the denominator) is 7, which is the length of the shortest repeating cycle.
• Solving Diophantine Equations: The HCF is used to solve Diophantine equations, which are equations involving integer variables. For example, the equation 3x + 5y = 1 can be solved using the HCF of 3 and 5, which is 1.

Common Mistakes to Avoid
When working with the HCF, it is important to avoid common mistakes, such as:

• Confusing HCF with LCM (Least Common Multiple): The HCF is the largest number that divides both numbers, while the LCM is the smallest number that is divisible by both numbers.
• Not considering negative numbers: The HCF is defined only for positive integers. If the numbers are negative, we must first find the HCF of their absolute values.
• Using a calculator: It is generally not necessary to use a calculator to find the HCF. Instead, prime factorization or the Euclidean algorithm can be used to find the HCF efficiently.

How to Find the HCF: Step-by-Step Approach
There are several methods to find the HCF of two numbers. One common method is to use the prime factorization method, which involves the following steps:

1. Prime factorize both numbers.
2. Identify the common prime factors.
3. Multiply the common prime factors together to get the HCF.

Additionally, the Euclidean algorithm can also be used to find the HCF. For two numbers a and b, the Euclidean algorithm works as follows:

1. Divide a by b and obtain the remainder r.
2. Divide b by r and obtain the remainder s.
3. Continue dividing until the remainder is 0.
4. The last non-zero remainder is the HCF of a and b.

FAQs
Q1. What is the difference between HCF and LCM?
A1. The HCF is the largest number that divides both numbers without leaving a remainder, while the LCM is the smallest number that is divisible by both numbers.

Q2. How can I find the HCF of three or more numbers?
A2. To find the HCF of three or more numbers, find the HCF of two numbers first and then find the HCF of the result with the third number. Continue this process until all the numbers have been considered.

Q3. Is the HCF of two consecutive numbers always 1?
A3. Yes, the HCF of two consecutive numbers is always 1.

Inspirational Stories

Story 1:
A farmer had 18 cows and 27 sheep. He wanted to divide them into equal groups so that each group had the same number of cows and sheep. What is the largest possible group size?

Answer: The largest possible group size is the HCF of 18 and 27, which is 9.

Lesson: The HCF can be used to solve practical problems involving divisibility.

Story 2:
A group of 6 friends wanted to take a picture together. They had two different cameras with 5 and 7 megapixel resolutions, respectively. What is the highest resolution at which they can all fit in a single picture?

Answer: The highest resolution is the HCF of 5 and 7, which is 1.

Lesson: The HCF can be used to determine the maximum compatibility between two or more different technologies.

Story 3:
A group of hikers had to cross a river using a boat that could carry at most 3 people at a time. They had 7 people in total. What is the minimum number of crossings needed for all of them to reach the other side?

Answer: The minimum number of crossings is the HCF of 7 and 3, which is 1.

Lesson: The HCF can be used to solve optimization problems involving divisibility.

Useful Tables