Coprime numbers: definition, examples and properties. “Greatest common divisor. Coprime numbers Learning new material in the form of a conversation
urban district of Tolyatti
“Greatest common divisor. Coprime numbers.
Teacher Kostina T.K.
g. o. Tolyatti
Presentation on theme: "Greatest Common Divisor.
Coprime Numbers"
Preliminary preparation for the lesson: students should know the following topics: "Divisors and multiples", "Signs of divisibility by 10, 5, 2, 3, 9", "Prime and composite numbers", "Decomposition into prime factors"
Lesson Objectives:
Educational: to study the concepts of GCD and relatively prime numbers; teach students to find GCD numbers; create conditions for developing the ability to summarize the studied material, analyze, compare and draw conclusions.
Educational: the formation of self-control skills; fostering a sense of responsibility.
Developing: development of memory, imagination, thinking, attention, ingenuity.
During the classes
Minutes of logical tasksOral work.
1. Grandparents brought an odd number of apricots from the garden for their two grandchildren. Can these apricots be divided equally among the grandchildren? [can]
2. From one village to another 3 km. Two people came out of these villages towards each other with the same speed. The meeting took place half an hour later. Find the speed of each.
3. The tourist has passed 2/5 of the whole way. After that, he had to go 4 km more than he did. Find all the way.
4. The number of eggs in the basket is less than 40. If they are counted in pairs, then 1 egg will remain. If you count them in triplets, then there will still be one egg each. How many eggs are in the basket? (31)
2. Repetition.
According to the table, we repeat the definition of a divisor, a multiple, signs of divisibility, the definition of prime and composite numbers. On the screen are slides depicting animals, a map of the Samara region, photographs of a VAZ.
3. Learning new material in the form of a conversation.
What are the divisors of the number 18, 21, 24.
The area of the VAZ is 500 hectares. Into what prime factors can this number be decomposed? 500=2*5*2*5*5=2 2 *5 3
What are the common divisors of the numbers 120 and 80.
The weight of the bear is 525 kg. The mass of an elephant is 5025 kg. Name some common divisors
The beaver weighs 24 kg and is 97 cm long. Which numbers are simple or complex? Name their common divisors.
56640 tons of oxygen is consumed by 1 passenger aircraft for 9 hours of operation. This amount of oxygen is released during photosynthesis of 35,000 hectares of forest. Name some divisors of this number.
Which of these numbers are prime and which are composite? 111, 313, 323, 437, 549, 677, 781, 891?
Which number is divisible by all numbers without a remainder?
What is the divisor of any natural number?
Is the expression 34*28+85*20 divisible by 17?
Is the expression 4132*7008 divisible by 3?
What is the quotient (3*5*2*7*13)/(5*2*13)=?
What is the product of (2*5*5*5*3)*(2*2*2*2*3)?
Name some prime numbers.
The further we go along the natural series of numbers, the more difficult it is to find prime numbers. Imagine that we are flying in an airplane that flies along a natural line. It's dark all around and only prime numbers are marked with lights. There are a lot of lights at the beginning of the journey, and then less and less.
The ancient Greek scientist Euclid proved 2300 years ago that there are infinitely many prime numbers and that there is no largest prime number.
The problem of prime numbers was studied by many mathematicians, including the ancient Greek scientist Eratosthenes. His method of finding prime numbers was called the sieve of Eratosthenes.
Goldbach and Euler, who lived in the 18th century and were members of the St. Petersburg Academy of Sciences, dealt with the problem of prime numbers. They assumed that every natural number can be represented as a sum of prime numbers, but this has not been proven. In 1937, the Soviet academician Vinogradov proved this proposition.
An Indian elephant lived for 65 years, a crocodile for 51 years, a camel for 23 years, and a horse for 19 years. Which of these numbers are prime and composite?
The wolf is chasing the hare, he needs to get through the labyrinth. You can pass if the answer is a prime number [mazes in the form of circles, on which there are three examples, and in the center there is a house]
1000-2; 250*2+9; 310/5
24/4, 2 2 +41, 23+140
10-3; 133+12; 28*5
To the task on the board record:
Divisors 48: 1, 2, 3, 4, 6, 8, 12, 16, 48
Divisors 36: 1, 2, 3, 4, 6, 12, 18, 36
GCD (48; 36) \u003d 12 12 gifts determination of the GCD of the divisor rule for finding the GCD
And how to find the GCD of large numbers, when it is difficult to list all the divisors. According to the table and the textbook, we derive the rule. We highlight the main words: decompose, compose, multiply.
I show examples of finding GCD from large numbers, here we can say that GCD of large numbers can be found using the Euclidean algorithm. We will get acquainted with this algorithm in detail in the classroom of the mathematical school.
An algorithm is a rule according to which actions are performed. In the 9th century, such rules were given by the Arab mathematician Alkhvaruimi.
4. Work in groups of 4 people.
Everyone gets one of 4 options for tasks, where the following is indicated:
The student must study the theory from the textbook and answer one question
Study an example of finding GCD
Complete tasks for independent work.
At the end of the work, a small independent work is carried out.
CSR cards
Option 1
1. What number is called prime? What is a composite number?
2. Find GCD (96; 36)
To find the GCD of numbers, you need to decompose the given numbers into prime factors.
| 96 | 2 |
| 48 | 2 |
| 24 | 2 |
| 12 | 2 |
| 6 | 2 |
| 3 | 3 |
| 1 |
| 36 | 2 |
| 18 | 2 |
| 9 | 3 |
| 3 | 3 |
| 1 | |
36=2 2 *3 2
96=2 5 *3
The expansion of the number that is the GCD of the numbers 96 and 36 will include the common prime factors with the smallest exponent:
GCD (96;36)=2 2 *3=4*3=12
3. Decide for yourself. GCD(102; 84), GCD(75; 28), GCD(120; 144)
Option 2
1. What does it mean to decompose a natural number into prime factors? What is the common divisor of these numbers?
2. Sample GCD (54; 72)=18
3. Solve yourself GCD(144; 128), GCD(81; 64), GCD(360; 840)
Option 3
1. What numbers are called relatively prime? Give an example.
2. Sample GCD (72; 96) =24
3. Solve yourself GCD(102; 170), GCD(45; 64), GCD(864; 192)
Option 4
1. How to find a common divisor of numbers?
2. Sample GCD (360; 432)
3. Solve yourself GCD (135; 105), GCD (128; 75), GCD (360; 8400)
Independent work
| Option 1 | Option 2 | Option 3 | Option 4 |
| NOD (180; 120) | NOD (150; 375) | NOD (135; 315; 450) | NOD (250; 125; 375) |
| NOD (2016; 1320) | NOD (504; 756) | NOD (1575, 6615) | NOD (468; 702) |
| NOD (3120; 900) | NOD (1028; 1152) | NOD (1512; 1008) | NOD (3375; 2250) |
5. Summing up the lesson. Reporting grades for independent work.
Identical gifts can be made from 48 Swallow sweets and 36 Cheburashka sweets, if you need to use all the sweets?
Solution. Each of the numbers 48 and 36 must be divisible by the number of gifts. Therefore, we first write out all the divisors of the number 48.
We get: 2, 3, 4, 6, 8, 12, 16, 24, 48.
Then we write out all the divisors of the number 36.
We get: 1, 2, 3, 4, 6, 9, 12, 18, 36.
The common divisors of the numbers 48 and 36 will be: 1, 2, 3, 4, 6, 12.
We see that the largest of these numbers is 12. It is called the greatest common divisor of the numbers 48 and 36.
So, you can make 12 gifts. Each gift will contain 4 "Swallow" sweets (48:12=4) and 3 "Cheburashka" sweets (36:12=3).
Lesson content lesson summary support frame lesson presentation accelerative methods interactive technologies Practice tasks and exercises self-examination workshops, trainings, cases, quests homework discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photos, pictures graphics, tables, schemes humor, anecdotes, jokes, comics, parables, sayings, crossword puzzles, quotes Add-ons abstracts articles chips for inquisitive cheat sheets textbooks basic and additional glossary of terms other Improving textbooks and lessonscorrecting errors in the textbook updating a fragment in the textbook elements of innovation in the lesson replacing obsolete knowledge with new ones Only for teachers perfect lessons calendar plan for the year methodological recommendations of the discussion program Integrated LessonsMath lesson in grade 5 A on the topic:
(according to the textbook by G.V. Dorofeev, L.G. Peterson)
Mathematics teacher: Danilova S.I.
Lesson topic: Greatest common divisor. Coprime numbers.
Lesson type: A lesson in learning new material.
The purpose of the lesson: Get a universal way to find the greatest common divisor of numbers. Learn how to find the GCD of numbers by factoring.
Formed results:
Subject: compose and master the algorithm for finding the GCD, train the ability to apply it in practice.
Personal: to form the ability to control the process and the result of educational and mathematical activities.
Metasubject: to form the ability to find the GCD of numbers, apply the signs of divisibility, build logical reasoning, inference and draw conclusions.
Planned results:
The student will learn how to find the GCD of numbers by factoring numbers into prime factors.
Basic concepts: GCD of numbers. Coprime numbers.
Forms of student work: frontal, individual.
Required technical equipment: teacher's computer, projector, interactive whiteboard.
Lesson structure.
Organizing time.
oral work. Gymnastics for the mind.
The topic of the lesson. Learning new material.
Fizkultminutka.
Primary consolidation of new material.
Independent work.
Homework. Reflection of activity.
During the classes
Organizing time.(1 minute.)
Stage tasks: to provide an environment for the work of class students and psychologically prepare them for communication in the upcoming lesson
Greetings:
Hello guys!
looked at each other,
And everyone quietly sat down.
The bell has already rung.
Let's start our lesson.
oral work. Mind gymnastics. (5 minutes.)
Tasks of the stage: recall and consolidate the algorithms for accelerated calculations, repeat the signs of divisibility of numbers.
In the old days in Russia they said that multiplication is torment, but trouble with division.
Anyone who could divide quickly and accurately was considered a great mathematician.
Let's see if you can be called great mathematicians.
Let's do mental gymnastics.
1) Choose from many
A=(716, 9012, 11211, 123400, 405405, 23025, 11175)
multiples of 2, multiples of 5, multiples of 3.
2) Calculate orally:
5 . 37 . 2 = 3. 50 . 12 . 3 . 2 =
2. 25 . 51 . 3 . 4 = 4. 8 . 125 . 7 =
Motivation for learning activities. Setting goals and objectives for the lesson.(4 min.)
Target :
1) the inclusion of students in educational activities;
2) organize the activities of students in setting the thematic framework: new ways of finding GCD numbers;
3) to create conditions for the emergence of the student's internal need for inclusion in educational activities.
– Guys, what topic did you work on in the last lessons? (On the decomposition of numbers into prime factors) What knowledge did we need in this case? (Signs of divisibility)
We opened the notebooks, let's check the home number number 638.
In your homework, you determined using factorization whether the number a is divisible by the number b and found the quotient. Let's check what you got. Checking #638. In which case is a divisible by b? If a is divisible by b, then what is b for a? What is b for a and b? And how do you think, how to find the GCD of numbers if one of them is not divisible by the other? What are your assumptions?
And now let's consider the problem: "What is the largest number of identical gifts that can be made from 48 "squirrel" candies and 36 "inspiration" chocolates, if you need to use all the candies and chocolates?"
Write on the board and in notebooks:
36=2*2*3*3
48=2*2*2*2*3
GCD(36,48)=2*2*3=12
How can we apply factorization to solve this problem? What do we actually find? GCD of numbers. What is the purpose of our lesson? Learn to find the GCD of numbers in a new way.
4. Post the topic of the lesson. Learning new material.(3.5 min.)
Write down the number and the topic of the lesson: Greatest Common Divisor.
(the greatest common divisor is the largest number that divides each of the given natural numbers). All natural numbers have at least one common divisor, 1.
However, many numbers have multiple common divisors. A universal way to search for GCD is to decompose these numbers into prime factors.
Let us write an algorithm for finding the GCD of several numbers.
Decompose these numbers into prime factors.
Find the same factors and underline them.
Find the product of common factors.
Physical education minute(get up from the desks) - flash video. (1.5 min.)
(Fallback:
We pulled up together
And they smiled at each other.
One - clap and two - clap.
Left foot - top, and right - top.
Shake your head -
Stretching the neck.
Top foot, now - another
We can do it all together.)
Primary consolidation of new material. ( 15 minutes. )
Implementation of the constructed project
Target:
1) organize the implementation of the constructed project in accordance with the plan;
2) organize the fixation of a new mode of action in speech;
3) organize the fixation of a new mode of action in signs (with the help of a standard);
4) organize the fixation of overcoming difficulties;
5) to organize a clarification of the general nature of the new knowledge (the possibility of applying a new method of action to solve all tasks of this type).
Organization of the educational process: № 650(1-3), 651(1-3)
№ 650 (1-3).
№ 650 (2) to disassemble in detail, because there are no common prime divisors.
The first point has been completed.
2. D (a; b) = no
3. GCD ( a; b ) = 1
What interesting things did you notice? (Numbers do not have common prime divisors.)
In mathematics, such numbers are called relatively prime numbers. Notebook entry:
Numbers whose greatest common divisor is 1 are called mutually simple.
a and b coprime gcd ( a ; b ) = 1
What can you say about the greatest common divisors of coprime numbers?
(The greatest common divisor of coprime numbers is 1.)
№ 651 (1-3)
The task is carried out at the blackboard with a commentary.
Let's decompose the numbers into prime factors using the well-known algorithm:
75 3 135 3
25 5 45 3
5 5 15 3
1 5 5
GCD (75; 135) \u003d 3 * 5 \u003d 15.
180 2*5 210 2*5
18 2 21 3
9 3 7 7
3 3 1
GCD (180, 210)=2*5*3=30
125 5 462 2
25 5 231 3
5 5 77 7
1 11 11
GCD (125, 462)=1
7. Independent work.(10 min.)
How to prove that you have learned to find the greatest common divisor of numbers in a new way? (You must do your own work.)
Independent work.
Find the greatest common divisor of numbers using prime factorization.
Option 1 Option 2
a=2 × 3 × 3 × 7 × 11 1) a=2 × 3 × 5 × 7 × 7
b=2×5×7×7×13 b=3×3×7×13×19
60 and 165 2) 75 and 135
81 and 125 3) 49 and 125
4) 180, 210 and 240 (optional)
Guys, try to apply your knowledge when doing independent work.
Students first do independent work, then peer-check and check with a sample on the slide.
Independent work check:
Option 1 Option 2
GCD(a,b)=2 × 7=14 1) GCD(a,b)=3 × 7=21
GCD( 60, 165 )=3 × 5 =15 2) GCD(75, 135)=3 × 5 =15
gcd(81, 125)=1 3) gcd(49, 125)=1
8. Reflection of activity.(5 minutes.)
What new did you learn in the lesson? (A new way to find the GCD using prime factors, which numbers are called coprime, how to find the GCD of numbers if a larger number is divisible by a smaller number.)
What was your goal?
Have you reached your goal?
What helped you achieve your goal?
– Determine the truth for yourself of one of the following statements (P-1).
What do you need to do at home to better understand this topic? (Read the paragraph, and practice finding the GCD with the new method).
Homework:
item 2, №№ 672 (1,2); 673 (1-3), 674.
Determine the truth for yourself of one of the following statements:
"I figured out how to find the GCD of numbers" 
"I know how to find the GCD of numbers, but I still make mistakes" 
"I have unanswered questions."
Display your answers as emojis on a piece of paper.
Finished works
THESE WORKS
Much is already behind and now you are a graduate, if, of course, you write your thesis on time. But life is such a thing that only now it becomes clear to you that, having ceased to be a student, you will lose all student joys, many of which you have not tried, putting everything off and putting it off for later. And now, instead of catching up, you're tinkering with your thesis? There is a great way out: download the thesis you need from our website - and you will instantly have a lot of free time!
Diploma works have been successfully defended in the leading Universities of the Republic of Kazakhstan.
Cost of work from 20 000 tenge
COURSE WORKS
The course project is the first serious practical work. It is with writing a term paper that preparation for the development of graduation projects begins. If a student learns to correctly state the content of the topic in a course project and correctly draw it up, then in the future he will have no problems either with writing reports, or with compiling theses, or with performing other practical tasks. In order to assist students in writing this type of student work and to clarify the questions that arise in the course of its preparation, in fact, this information section was created.
Cost of work from 2 500 tenge
MASTER'S THESES
At present, in higher educational institutions of Kazakhstan and the CIS countries, the stage of higher professional education, which follows after the bachelor's degree - the master's degree, is very common. In the magistracy, students study with the aim of obtaining a master's degree, which is recognized in most countries of the world more than a bachelor's degree, and is also recognized by foreign employers. The result of training in the magistracy is the defense of a master's thesis.
We will provide you with up-to-date analytical and textual material, the price includes 2 scientific articles and an abstract.
Cost of work from 35 000 tenge
PRACTICE REPORTS
After completing any type of student practice (educational, industrial, undergraduate) a report is required. This document will be a confirmation of the student's practical work and the basis for the formation of the assessment for the practice. Usually, in order to compile an internship report, you need to collect and analyze information about the enterprise, consider the structure and work schedule of the organization in which the internship takes place, draw up a calendar plan and describe your practical activities.
We will help you write a report on the internship, taking into account the specifics of the activities of a particular enterprise.
Common divisors
Example 1
Find the common divisors of the numbers $15$ and $–25$.
Solution.
Divisors of the number $15: 1, 3, 5, 15$ and their opposites.
Divisors of the number $–25: $1, $5, $25 and their opposites.
Answer: $15$ and $–25$ have common divisors of $1, 5$ and their opposites.
According to the divisibility properties, the numbers $−1$ and $1$ are divisors of any integer, so $−1$ and $1$ will always be common divisors for any integers.
Any set of integers will always have at least $2$ common divisors: $1$ and $−1$.
Note that if the integer $a$ is a common divisor of some integers, then -a will also be a common divisor of those integers.
Most often, in practice, they are limited only to positive divisors, but do not forget that each integer opposite to a positive divisor will also be a divisor of this number.
Finding the Greatest Common Divisor (GCD)
According to the properties of divisibility, every integer has at least one divisor other than zero, and the number of such divisors is finite. In this case, the common divisors of the given numbers are also a finite number. Of all the common divisors of given numbers, you can select the largest number.
If all these numbers are equal to zero, it is impossible to determine the largest of the common divisors, because zero is divisible by any integer, of which there are an infinite number.
The greatest common divisor of numbers $a$ and $b$ in mathematics is denoted as $gcd(a, b)$.
Example 2
Find the gcd of the integers 412$ and $–30$..
Solution.
Let's find the divisors of each of the numbers:
$12$: numbers $1, 3, 4, 6, 12$ and their opposites.
$–30$: numbers $1, 2, 3, 5, 6, 10, 15, 30$ and their opposites.
The common divisors of the numbers $12$ and $–30$ are $1, 3, 6$ and their opposites.
$gcd (12, -30)=6$.
It is possible to determine the GCD of three or more integers in the same way as the definition of the GCD of two numbers.
GCD of three or more integers is the largest integer that divides all numbers simultaneously.
Denote the largest divisor $n$ of numbers $gcd(a_1, a_2, …, a_n)= b$.
Example 3
Find the gcd of three integers $–12, 32, 56$.
Solution.
Let's find all divisors of each of the numbers:
$–12$: numbers $1, 2, 3, 4, 6, 12$ and their opposites;
$32$: numbers $1, 2, 4, 8, 16, 32$ and their opposites;
$56$: numbers $1, 2, 4, 7, 8, 14, 28, 56$ and their opposites.
The common divisors of the numbers $–12, 32, 56$ are $1, 2, 4$ and their opposites.
Find the largest of these numbers by comparing only the positive ones: $1
$gcd(-12, 32, 56)=4$.
In some cases the gcd of integers can be one of these numbers.
Coprime numbers
Definition 3
Integers $a$ and $b$ – coprime, if $gcd(a, b)=1$.
Example 4
Show that the numbers $7$ and $13$ are coprime.
