The highest common factor (HCF) of 48 and 80 is 16.
Understanding the Highest Common Factor (HCF)
The Highest Common Factor (HCF), often referred to as the Greatest Common Divisor (GCD), is the largest positive integer that divides two or more numbers without leaving a remainder. It's a fundamental concept in mathematics that helps in simplifying fractions and solving various number theory problems. Understanding how to find the HCF can be incredibly useful in both academic settings and practical applications.
To calculate the HCF of two numbers, such as 48 and 80, we need to identify all the factors of each number and then find the largest factor they share.
Steps to Determine the HCF of 48 and 80
Let's break down the process using the numbers 48 and 80:
-
List the Factors of Each Number:
- Factors are numbers that divide a given number exactly, without any remainder.
- For 48, the factors are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
- For 80, the factors are: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80.
We can present these factors clearly in a table:
Number Factors 48 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 80 1, 2, 4, 5, 8, 10, 16, 20, 40, 80 -
Identify Common Factors:
- Next, we look for the factors that appear in both lists. These are called common factors.
- The common factors of 48 and 80 are: 1, 2, 4, 8, and 16.
-
Select the Highest Common Factor:
- From the list of common factors, the highest (greatest) one is the HCF.
- In this case, the greatest factor that exactly divides both 48 and 80 is 16.
Why is HCF Important?
The HCF is crucial in several mathematical contexts:
- Simplifying Fractions: To simplify a fraction to its lowest terms, you divide both the numerator and the denominator by their HCF. For example, to simplify 48/80, you would divide both by 16, resulting in 3/5.
- Distributing Items: Imagine you have 48 apples and 80 oranges, and you want to create the largest possible number of identical fruit baskets without any fruit left over. The HCF (16) tells you that you can make 16 baskets, each with 3 apples (48/16) and 5 oranges (80/16).
- Solving Word Problems: Many real-world problems involve finding the largest group or measure that can be made from given quantities, which directly relates to the HCF.
For more information on HCF and how to calculate it using different methods (like prime factorization), you can refer to educational resources such as Khan Academy's explanation of GCF.
