Greatest Common Divisor (gcd)
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Understanding Divisors
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Today we are going to talk about divisors. Can anyone tell me what a divisor is?
A divisor is a number that divides another number perfectly without a remainder.
Great! Now, if we have two numbers, how can we find their greatest common divisor, or gcd?
I think we have to list out all the divisors of each number, right?
Exactly! We will then look for the largest number that appears in both lists. Let’s take 8 and 12 as an example. What do you think their gcd is?
I believe the gcd is 4 since it is the highest common divisor!
Absolutely, well done! Remember, gcd stands for greatest common divisor and gives us the largest number that divides both inputs.
Algorithms for Finding gcd
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Now that we understand divisors, let's discuss how we can systematically compute the gcd. What’s our initial step?
We start by listing all the factors of the first number.
Correct! Then what do we do with the second number?
List its divisors too, and then find the common divisors between the two lists.
Right! Finally, how do we determine the gcd from our common divisors?
We find the largest one among them!
Exactly! This algorithm is not the most efficient, but it's a clear way to understand the process.
Working Example of gcd
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
Let's work through an example: What is the gcd of 14 and 63? What are the first steps?
First, we list the factors of 14, which are 1, 2, 7, and 14.
Great! Now let’s move on to 63.
The factors of 63 are 1, 3, 7, 9, 21, and 63.
Good job! Now, what are the common factors between 14 and 63?
The common factors are 1 and 7!
Exactly! Therefore, the gcd of 14 and 63 is 7. Excellent teamwork!
Efficiency in Algorithms
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
While we have learned a basic algorithm for computing gcd, are there more efficient ways we could approach this?
I think we can use something like the Euclidean algorithm?
Exactly! The Euclidean algorithm provides a much faster way to compute the gcd without listing factors. We will cover this method next time!
Recap and Application
🔒 Unlock Audio Lesson
Sign up and enroll to listen to this audio lesson
To wrap this up, can anyone give a brief summary of what we've learned about the gcd?
We've learned that the gcd is the largest divisor shared by two numbers and how to compute it using an algorithm!
Plus, we worked through examples to solidify our understanding.
Wonderful! Remember, understanding how to find the gcd is important in many areas of mathematics, including number theory and cryptography.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section explains what a gcd is and provides methods for computing it through an algorithmic approach. It includes examples to illustrate the results, such as the gcd of two integers and an illustrative procedure for listing factors.
Detailed
Greatest Common Divisor (gcd)
In this section, we explore the concept of the Greatest Common Divisor (gcd) of two positive integers, defined as the largest integer that divides both numbers without leaving a remainder. The section begins with foundational definitions, including the meaning of a divisor and how to identify common divisors.
Key Concepts:
- Divisor: A number that divides another number without leaving a remainder.
- Greatest Common Divisor (gcd): The largest number that divides two integers, which can be computed using various algorithms.
Example of gcd:
- For the integers 8 and 12, common divisors include 1, 2, 4, and 8; thus, gcd(8, 12) = 4. Conversely, gcd(18, 25) = 1, as their only common divisor is 1.
Basic Algorithm for Computing gcd:
- List all factors of both integers.
- Identify common factors between the two lists.
- Return the largest common divisor as the gcd.
This algorithm highlights the systematic approach to computing the gcd using simple steps and emphasizes understanding how to efficiently determine factors.
Youtube Videos
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Understanding the Greatest Common Divisor
Chapter 1 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
To illustrate this let us look at the function which most of us have seen and try to understand the algorithmically. So, the property that I want to compute is the greatest common decide divisor of two positive integers m and n. So, as we know a divisor is a number that divides. So k is a divisor of m; if I can divide m by k and get no reminder. So, the greatest common divisor of m and n must be something which is a common divisor. So, common means it must divide both and it must be the largest of these. So, if the largest k suchthatk divides mand k also divides m.
Detailed Explanation
The greatest common divisor, or gcd, of two numbers is the largest number that can evenly divide both numbers without leaving a remainder. For example, if we take m = 8 and n = 12, the divisors of 8 are 1, 2, 4, and 8, while the divisors of 12 are 1, 2, 3, 4, 6, and 12. The common divisors are 1, 2, and 4. Among these, the greatest is 4, so we say that gcd(8, 12) = 4. Similarly, for m = 18 and n = 25, the only common divisor is 1, which means gcd(18, 25) = 1.
Examples & Analogies
Imagine you have 8 apples and 12 oranges. You want to divide them into baskets such that each basket has an equal quantity of both fruits without cutting them. The largest number of baskets you can create with the same number of apples and oranges in each without leftovers would represent the gcd, which in this case is 4.
Computing the gcd
Chapter 2 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
So, how would one go about computing gcd of m, n? So, this is where we come to the algorithmic way, we want to describe the uniform way of systematically computing gcd of m n for any m or any n. So, here is a very simple procedure. It is not the most efficient; we will see better once as we go along. But if we just look at the definition of gcd it says look at all the factors of m, look at all the factor of n and find the largest one which is the factor of both.
Detailed Explanation
To compute the gcd of two numbers, m and n, we can follow these steps: First, list all the factors of m. Then, list all the factors of n. Finally, identify the largest number that appears in both lists. For instance, if m = 14 and n = 63, we find the factors of 14 (1, 2, 7, 14) and the factors of 63 (1, 3, 7, 9, 21, 63). The common factors are 1 and 7. Thus, gcd(14, 63) = 7.
Examples & Analogies
Think of two teams of kids preparing for a relay race. One team can form groups of 14 members, while the other forms groups of 63. To find out the largest identical team size they can form to compete together without anyone left out, they need to find the gcd of 14 and 63, which is 7.
Listing Factors Algorithmically
Chapter 3 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
To argue that this process is well defined all we need to realize is that the factors of m must be between 1 and m. In other words, we although there are infinitely many different possibilities as factors we don't have to look at any number bigger than m, because it cannot go into m evenly.
Detailed Explanation
The factors of a number m can only be between 1 and m itself. By systematically checking each number between 1 and m, we can determine if a number is a factor by checking if m divided by that number has a remainder of 0. This means we only need to check numbers in that range, efficiently narrowing down our possibilities.
Examples & Analogies
Imagine you have a jar containing 14 candies. The only possible ways to divide those candies into groups are by having 1 to 14 candies in each group. You only need to check those amounts to see how many complete groups you can create without leftovers.
Finding Common Factors
Chapter 4 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
Having computed the factors of the two numbers 14 and 63, the next step in our algorithm says that we must find the largest factor that appears in both list.
Detailed Explanation
Once we have the lists of factors for both numbers, we need to compare them to find common factors. In our example, the factors of 14 are [1, 2, 7, 14] and those of 63 are [1, 3, 7, 9, 21, 63]. The common factors are 1 and 7. The greatest common factor is thus 7, which is simply the largest number that appears in both lists.
Examples & Analogies
If the kids in the previous example each received different amounts of candy, after they counted how many they had, they could only share the candies that they both had in common. Thus, they realize they can each make groups of 7 candies to share with each other.
Conclusion of the gcd Algorithm
Chapter 5 of 5
🔒 Unlock Audio Chapter
Sign up and enroll to access the full audio experience
Chapter Content
And now having done this we have a list of all the common factors we computed them from smallest to biggest, because we went to the factors of 14 in ascending order.
Detailed Explanation
By systematically comparing the factors from both lists in ascending order, we can ensure that the final result, the gcd, is the highest number that both numbers share. This method guarantees we won't miss any common factor, and we are assured the result is comprehensive and correct.
Examples & Analogies
Returning to our candy example, after making sure they only divided what they could share, they can proudly show that the biggest group they could have together is aimed specifically at the size of 7 candies.
Key Concepts
-
Divisor: A number that divides another number without leaving a remainder.
-
Greatest Common Divisor (gcd): The largest number that divides two integers, which can be computed using various algorithms.
-
Example of gcd:
-
For the integers 8 and 12, common divisors include 1, 2, 4, and 8; thus, gcd(8, 12) = 4. Conversely, gcd(18, 25) = 1, as their only common divisor is 1.
-
Basic Algorithm for Computing gcd:
-
List all factors of both integers.
-
Identify common factors between the two lists.
-
Return the largest common divisor as the gcd.
-
This algorithm highlights the systematic approach to computing the gcd using simple steps and emphasizes understanding how to efficiently determine factors.
Examples & Applications
The gcd of 8 and 12 is 4.
The gcd of 18 and 25 is 1.
The gcd of 14 and 63 is 7.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
Divisor divides, cancels and correlates, to find the gcd, we celebrate!
Stories
Once in a village, there lived two traders, each carrying bags of goods. Every time they met, they shared their goods fairly, the largest number they both could share was the gcd of their goods.
Memory Tools
To find gcd, remember: 'List, Match, Max!' It’s about listing factors, matching them, and finding the maximum.
Acronyms
GCD
Great Common Divide
Flash Cards
Glossary
- Divisor
A number that divides another number without leaving a remainder.
- Greatest Common Divisor (gcd)
The largest positive integer that can divide two or more integers without leaving a remainder.
- Algorithm
A step-by-step procedure or formula for solving a problem.
- Common Factors
Factors that are shared by two or more numbers.
Reference links
Supplementary resources to enhance your learning experience.