Computing the gcd of Two Numbers
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Introduction to gcd
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Today, we're going to explore the concept of the greatest common divisor, simply known as gcd. Can anyone tell me what they think it is?
Is it the largest number that can divide two other numbers without leaving a remainder?
Exactly! The gcd is the largest number that divides both without a remainder. For instance, if we take 8 and 12, what do you think their gcd might be?
It's 4, right? Because both 8 and 12 can be divided by 4!
That's right! Remember, when we’re finding the gcd, we need to identify all the divisors first. A good acronym to remember the key steps is GCD: *Gather, Compare, Decide.*
So, we gather the factors, compare them, and then decide on the largest one!
Perfect! Let's explore how to compute it using an algorithm.
Calculating factors
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First, we need to compute the factors of the numbers we're comparing. How would we find the factors of 14?
We can check each number from 1 to 14 to see if it divides 14 without a remainder.
Exactly! When you check each number, what are the factors you find?
The factors of 14 are 1, 2, 7, and 14.
Correct! Now, how about the factors of 63?
They are 1, 3, 7, 9, 21, and 63.
Good job! Now that we have both lists, can someone summarize the next step?
We need to find the common factors from both lists.
Correct! Remember the phrase *Common Connect* to help remember this step. Let's do it!
Finding common factors
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Now, how do we find common factors between the two sets we identified?
We can compare each factor of 14 to the factors of 63.
Yes! Which common factors do we find?
The common factors are 1 and 7.
And from those common factors, how do we determine the gcd?
We choose the largest one, which is 7!
Exactly! That’s how you compute the gcd efficiently through comparison. Keep practicing!
Applying the gcd algorithm
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Let's put our knowledge to test! What is the gcd of 18 and 25?
The factors of 18 are 1, 2, 3, 6, 9, and 18, while the factors of 25 are 1, 5, and 25.
Great! Now, what are the common factors?
The only common factor is 1, so the gcd is 1.
Correct! This is an excellent example of how gcd can be applied. For all pairs of positive integers, the gcd is always defined.
So, every pair will at least have a gcd of 1?
Right! Always keep this in mind when working with gcd.
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section discusses the definition of the greatest common divisor (gcd) and provides a detailed explanation of a naive algorithm for calculating the gcd of two numbers. The algorithm involves listing factors and finding the largest common divisor, along with practical examples to facilitate understanding.
Detailed
Detailed Summary
In this section, we delve into the concept of the greatest common divisor (gcd), which is the largest number that can evenly divide two given integers, m and n. To illustrate, if we take the numbers 8 and 12, their divisors are calculated to show that the gcd is 4, as it is the largest number that divides both. The greatest common divisor is significant as it underscores relationships among numbers and has practical applications in areas such as fractions and number theory.
The algorithm introduced for finding the gcd is straightforward, albeit naive: first, generate the list of factors for both numbers and then identify the largest number common to both lists. This simple procedure ensures clarity, although it is not the most efficient method for larger numbers. The necessity for an efficient algorithm emerges as the section progresses. Key steps include:\n1. Listing all factors of the first number, m, by checking each number from 1 to m to see if it divides m without remainder.
2. Repeating this step for the second number, n.
3. Finally, checking common factors from both lists and returning the largest one as the gcd.
Through examples of computing gcd for various number pairs, such as 14 and 63, the algorithm's mechanics are showcased. The section sets a foundational understanding of computational algorithms while preparing for more complex methods of determining the gcd later in the course.
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Understanding GCD
Chapter 1 of 4
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Chapter Content
So 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 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.
Detailed Explanation
The Greatest Common Divisor (GCD) refers to the largest positive integer that divides both numbers without leaving a remainder. For example, if we take the numbers 8 and 12, their divisors are 1, 2, 4, and 8 for 8, and 1, 2, 3, 4, 6, and 12 for 12. The largest number that appears in both lists is 4, which means GCD(8, 12) = 4.
Examples & Analogies
Think of the GCD like finding the largest piece of fabric you can cut into smaller pieces without any leftover scraps. If one piece is 8 inches wide and another is 12 inches wide, you can cut them both into pieces 4 inches wide perfectly.
Examples of GCD
Chapter 2 of 4
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Chapter Content
For instance, if we look at the number 8 and 12, then we can see that 4 is the factor of 8, 4 is the divisor of 8, 4 is also divisor of 12, another divisor of 12 is 6, but 6 is not a divisor of 8.
Detailed Explanation
When examining two numbers, 8 and 12, we find their common divisors. 4 divides both, making it the GCD. A second example shows that for 18 and 25, their only common divisor is 1, making GCD(18, 25) = 1. This showcases the process of determining if any numbers divide both given integers evenly.
Examples & Analogies
Imagine you are trying to find the largest size of pizza you can equally share between friends with toppings that don't exceed anything above a certain amount. The common divisor reflects how many pizzas can be made with each friend getting an equal share.
Algorithmic Approach to GCD
Chapter 3 of 4
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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.
Detailed Explanation
The naive method for calculating GCD involves listing factors of both numbers and identifying the largest common factor. Firstly, we list all divisors (factors) for both m and n. Next, we compare these lists and find the highest number that is common.
Examples & Analogies
This process can be related to sorting items by their sizes in order to find the biggest container to hold all of them without any space wasted. Just like checking each possible container size methodically, we check factors one by one.
Finding Common Factors
Chapter 4 of 4
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Chapter Content
Now having done this we have identified that factors of 14 and these factors are precisely the 1, 2, 7 and14. ...So, if we go through this systematically from one to 63 crossing out each number which is not a factor we end up with the list 1, 3, 7, 9, 21 and 63.
Detailed Explanation
After calculating common factors for 14 and 63, we note 1, 2, 7, 14 as factors of 14, and 1, 3, 7, 9, 21, 63 for 63. Common factors between both are determined by checking from one list against the other, revealing common elements such as 1 and 7.
Examples & Analogies
Imagine two teams with different sizes going to play in a tournament. You can only form groups of players of equal size. Finding the largest size of the group that can be formed by both teams helps determine how best to organize their participation.
Key Concepts
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Greatest Common Divisor (gcd): The largest integer that divides two numbers without a remainder.
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Factors: Numbers that divide another number evenly.
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Algorithm: A systematic procedure for calculations.
Examples & Applications
For the numbers 18 and 25, the gcd is 1 because their only common divisor is 1.
For the numbers 14 and 63, the gcd is 7 because both share the divisors 1 and 7.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
Gcd is the key, for numbers we see; the highest they share, gives harmony!
Stories
Imagine two friends trying to divide a pizza. They find the largest slice they can both share equally, and that slice size is their gcd!
Memory Tools
GCD = Gather, Compare, Decide - first gather the factors, compare them, and decide the greatest!
Acronyms
Use GCD as a reminder
Gather the factors
Check for common
Divide to find the greatest!
Flash Cards
Glossary
- Greatest Common Divisor (gcd)
The largest positive integer that divides two or more integers without leaving a remainder.
- Factor
A number that divides another number evenly without leaving a remainder.
- Divisor
A number by which another number is to be divided.
Reference links
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