Problems on the topic Greatest Common Divisor. Coprime numbers. “Greatest common divisor. Coprime Numbers Topic Greatest Common Divisor of Coprime Numbers

Math 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: largest 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 min.)

The tasks of the stage: to provide an environment for the work of the students in the class 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) inclusion of students in learning 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?

Now let's consider the problem: "What the largest number can you make identical gifts out of 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.

(greatest common divisor is largest number, by which each of the data is divided 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 educational process: № 650(1-3), 651(1-3)

№ 650 (1-3).

№ 650 (2) to disassemble in detail, because general prime divisors no.

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 minutes.)

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.

09.07.2015 6119 0

Goals: to form the skill of finding the greatest common divisor; introduce the concept of relatively prime numbers; to develop the ability to solve problems on the use of GCD numbers; learn to analyze, draw conclusions.

II. Verbal counting

1. Can the prime factorization of 24753 contain a factor of 5? Why? (No, because this number does not end with a 0 or 5.)

2. Name a number that is divisible by all numbers without a remainder. (Zero.)

3. The sum of two integers is odd. Is their product even or odd? (If the sum of two numbers is odd, then one number is even, the second is odd. Since one of the factors even number, therefore, it is divisible by 2, which means that the product is also divisible by 2. Then the whole product is even.)

4. In one family, each of the three brothers has a sister. How many children are in the family? (4 children: three boys and one sister.)

III . Individual work

Expand the number 210 in every possible way:

a) by 2 multipliers; (210 = 21 10 = 14 15 = 7 30 = 70 3 = 6 35 = 42 5 = 105 2.)

b) by 3 multipliers; (210 = 3 7 10 = 5 3 14 = 7 5 6 = 35 2 3 = 21 2 5 = 7 2 15.)

c) by 4 multipliers. (210 = 3 7 2 5.)

IV. Lesson topic message

"Numbers rule the world." These words belong to the ancient Greek mathematician Pythagoras, who lived in the 5th century. BC.

Today we will get acquainted with another group of numbers, which are called coprime.

V. Learning new material

1. Preparatory work.

No. 146 p. 25 (on the board and in notebooks). (On their own, at this time one student is working on reverse side boards.)

Find all divisors of each number.

Underline their common divisors.

Write down the greatest common divisor.

Answer:

What numbers have only one common divisor? (35 and 88.)

2. Work on a new theme.

(On their own, at this time one student works on the back of the board.)

Find the greatest common divisor of numbers: 7 and 21; 25 and 9; 8 and 12; 5 and 3; 15 and 40; 7 and 8.

Answer:

GCD (7; 21) = 7; GCD (25; 9) = 1; GCD (8; 12) = 4;

GCD (5; 3)= 1; GCD (15; 40) = 5; GCD (7; 8) = 1.

What pairs of numbers have the same common divisor? (25 and 9; 5 and 3; 7 and 8 is a common divisor of 1.)

Such numbers are called relatively prime.

Define relatively prime numbers.

Give examples of relatively prime numbers. (35 and 88, 3 and 7; 12 and 35; 16 and 9.)

VI. Historical moment

The ancient Greeks came up with a wonderful way to find the greatest common divisor of two natural numbers without factoring. It was called "Euclid's Algorithm".

About the life of the Greek mathematician Euclid, reliable data are unknown. He owns an outstanding scientific work called "Beginnings". It consists of 13 books and lays out the foundations of all ancient Greek mathematics.

It is here that Euclid's algorithm is described, which lies in the fact that the greatest common divisor of two natural numbers is the last one, which is different from zero, the remainder when these numbers are successively divided. Sequential division means division more to a smaller number, a smaller number to the first remainder, the first remainder to the second remainder, etc., until the division ends without a remainder. Suppose we need to find GCD (455; 312), then

455: 312 = 1 (rest. 143), we get 455 = 312 1 + 143.

312: 143 = 2 (rest. 26), 312 = 143 2 + 26,

143: 26 = 5 (rest 13), 143 = 26 5 + 13,

26: 13 = 2 (remaining 0), 26 = 13 2.

The last divisor or the last non-zero remainder is 13 and will be the required gcd (455; 312) = 13.

VII. Physical education minute

VIII. Working on a task

1. No. 152, p. 26 (with detailed commentary at the blackboard and in notebooks).

Read the task.

What is the task about?

What is the task about?

Name the 1st question of the task.

How to find out how many children were on the Christmas tree? (Find the GCD of numbers 123 and 82.)

Read the assignment for this task from the notebooks. (The number of oranges and apples must be divisible by the same largest number.)

How to find out how many oranges were in each gift? (Divide the entire number of oranges by the number of children present at the tree.)

How to find out how many apples were in each gift? (Divide the entire number of apples by the number of children present at the tree.)

Write down the solution of the problem in notebooks on a printed basis.

Decision:

GCD (123; 82) \u003d 41, which means 41 people.

123:41 = 3 (ap.)

82:41 = 2 (apple)

(Answer: 41 guys, 3 oranges, 2 apples.)

2. No. 164 (2) p. 27 (after a brief analysis, one student is on the back of the board, the rest on their own, then self-examination).

Read the task.

What is the degree measure of a straightened angle?

If one angle is 4 times smaller, then what about the second angle? (He's 4 times bigger.)

Write it down in a short note.

How will you solve the problem? (Algebraic.)

Decision:

1) Let x be the degree measure of the angle SOK,

4x - degree measure of an angle COD.

Since the sum of the angles SOC and COD equals 180°, then we write the equation:

x + 4x = 180

5x = 180

x=180:5

x = 36; 36° - degree measure of the SOC angle.

2) 36 4 \u003d 144 ° - degree measure of the angle COD.

(Answer: 36°, 144°.)

Build those corners.

Determine the type of angles SOK and COD . (Angle SOK - acute, angle KOD - dumb.)

Why?

IX. Consolidation of the studied material

1. No. 149 p. 26 (at the board with a detailed commentary).

What needs to be done to determine if the numbers are coprime? (Find their greatest common divisor, if it is equal to 1, then the numbers are coprime.)

2. No. 150 p. 26 (oral).

Validate your answer. (9 and 14; 14 and 15; 14 and 27 are pairs of relatively prime numbers, since their gcd is 1.)

3. No. 151 p. 26 (one student at the blackboard, the rest in notebooks).

(Answer: .)

Who disagrees?

4. Orally, with a detailed explanation.

How to find the greatest common divisor of several natural numbers? (Find in the same way as two numbers.)

Find the greatest common divisor of numbers:

a) 18, 14 and 6; b) 26, 15 and 9; c) 12, 24, 48; d) 30, 50, 70.

Decision:

a) 1. Check if the numbers 18 and 14 are divisible by 6. No.

2. We factorize the smallest number 6 = 2 3 into prime factors.

3. Check if the numbers 18 and 14 are divisible by 3. No.

4. Check if the numbers 18 and 14 are divisible by 2. Yes. Therefore, gcd (18; 14; 6) = 2.

b) GCD (26; 15; 9) = 1.

What can be said about these numbers? (They are relatively prime.)

c) GCD (12; 24; 48) = 12.

d) GCD (30; 50; 70) = 10.

X. Independent work

Mutual verification. (Answers are written on the closing board.)

Option I. No. 161 (a, b) p. 27, No. 157 (b - 1 and 3 numbers) p. 27.

Option II . No. 161 (c, d) p. 27, No. 157 (b - 2nd and 3rd number) p. 27.

XI. Summing up the lesson

What numbers are called coprime?

How can you find out if the given numbers are coprime?

How to find the greatest common divisor of several natural numbers?

Homework

No. 169 (6), 170 (c, d), 171, 174 p. 28.

Additional task:When you rearrange the digits of the prime number 311, you again get a prime number (check this on the table of prime numbers). Find all two-digit numbers that have the same property. (113, 131; 13, 31; 17, 71; 37, 73; 79, 97.)

Identical gifts can be made from 48 Swallow sweets and 36 Cheburashka sweets, if you need to use all the sweets?

Decision. 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 photographs, 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 guidelines discussion programs Integrated Lessons

Prime and Composite Numbers

Definition 1 . The common divisor of several natural numbers is the number that is a divisor of each of these numbers.

Definition 2 . The largest common divisor is called greatest common divisor (gcd).

Example 1 . The common divisors of the numbers 30 , 45 and 60 will be the numbers 3 , 5 , 15 . The greatest common divisor of these numbers will be

gcd(30, 45, 10) = 15.

Definition 3 . If the greatest common divisor of several numbers is 1, then these numbers are called coprime.

Example 2 . The numbers 40 and 3 will be coprime, but the numbers 56 and 21 are not coprime because the numbers 56 and 21 have a common divisor 7 which is greater than 1.

Remark . If the numerator of a fraction and the denominator of a fraction are relatively prime numbers, then such a fraction is irreducible.

Algorithm for Finding the Greatest Common Divisor

Consider algorithm for finding the greatest common divisor several numbers in the following example.

Example 3 . Find the greatest common divisor of the numbers 100, 750 and 800 .

Decision . Let's decompose these numbers into prime factors:

The prime factor 2 enters the first factorization to the power of 2, the second factorization to the power of 1, and the third factorization to the power of 5. Denote least of these degrees with the letter a. It's obvious that a = 1 .

The prime factor 3 enters the first factorization to the power of 0 (in other words, the factor 3 does not enter the first factorization at all), the second factorization enters the power of 1, and the third factorization to the power of 0. Denote least of these degrees with the letter b. It's obvious that b = 0 .

The prime factor 5 enters the first factorization to the power of 2, the second factorization to the power of 3, and the third factorization to the power of 2. Denote least of these degrees by the letter c. It's obvious that c = 2 .

Competition for young teachers

Bryansk region

"Pedagogical debut - 2014"

2014-2015 academic year

Math consolidation lesson in grade 6

on the topic "NOD. Coprime Numbers"

Place of work:MBOU "Glinishchevskaya secondary school" of the Bryansk region

Goals:

Educational:

• Consolidate and systematize the studied material;
• To develop the skills of decomposing numbers into prime factors and finding the GCD;
• Check students' knowledge and identify gaps;

Developing:

• Contribute to the development of students' logical thinking, speech and skills of mental operations;
• To contribute to the formation of the ability to notice patterns;
• Contribute to raising the level of mathematical culture;

Educational:

• To promote the formation of interest in mathematics; the ability to express one's thoughts, listen to others, defend one's point of view;
• education of independence, concentration, concentration of attention;
• to instill the skills of accuracy in keeping a notebook.

Lesson type: lesson of generalization and systematization of knowledge.

Teaching methods : explanatory and illustrative, independent work.

Equipment: computer, screen, presentation, handout.

During the classes:

1. Organizing time.

“The bell rang and fell silent - the lesson begins.

You quietly sat down at your desks, everyone looked at me.

Wish each other success with your eyes.

And forward for new knowledge.

Friends, on the tables you see the “Evaluation Sheet”, i.e. in addition to my evaluation, you will evaluate yourself by completing each task.

Evaluation paper

Guys, what topic did you study for several lessons? (We learned to find the greatest common divisor).

What do you think we will do today? State the topic of our lesson. (Today we will continue working with the greatest common divisor. The topic of our lesson is “The greatest common divisor”. In this lesson, we will find the greatest common divisor of several numbers, and solve problems using the knowledge of finding the greatest common divisor.).

Open your notebooks, write down the number, Classwork and the topic of the lesson: “Greatest common divisor. Coprime numbers.

1. Knowledge update

Several theoretical questions

Are the statements true? "Yes" - __; "No" - /\. slide 3-4

• A prime number has exactly two divisors; (right)
• 1 is a prime number; (not true)
• The smallest two-digit prime number is 11; (right)
• The largest two-digit composite number is 99; (right)
• The numbers 8 and 10 are coprime (not true)
• Some composite numbers cannot be decomposed into prime factors; (not true).

Key: _ /\ _ _/\ /\.

Evaluated their oral work in the evaluation sheet.

1. Systematization of knowledge

Today in our lesson there will be a little magic.

Where is the magic found? (in a fairy tale)

Guess from the picture what kind of fairy tale we will fall into. ( slide 5 ) Fairy tale Geese-swans. Absolutely right. Well done. And now let's all together try to remember the content of this tale. The chain is very short.

There lived a man and a woman. They had a daughter and a little son. Father and mother went to work and asked their daughter to look after her brother.

She put her brother on the grass under the window, and she ran out into the street, played, took a walk. When the girl returned, her brother was gone. She began to look for him, she screamed, called him, but no one answered. She ran out into an open field and only saw: swan geese rushed in the distance and disappeared behind a dark forest. Then the girl realized that they had taken away her brother. She had known for a long time that swan geese carried off small children.

She rushed after them. On the way, she met a stove, an apple tree, a river. But our river is not milky in the jelly banks, but an ordinary one, in which there are very, very many fish. None of them suggested where the geese flew, because she herself did not fulfill their requests.

For a long time the girl ran through the fields, through the forests. The day is already drawing to a close, suddenly she sees - there is a hut on a chicken leg, with one window, it turns around itself. In the hut, the old Baba Yaga is spinning a tow. And her brother is sitting on a bench by the window. The girl did not say that she had come for her brother, but lied, saying that she was lost. If it were not for the little mouse that she fed with porridge, then Baba Yaga would have fried it in the oven and eaten it. The girl quickly grabbed her brother and ran home. Geese - swans noticed them and flew after them. And whether they get home safely - everything now depends on us guys. Let's continue the story.

They run and run and run to the river. They asked to help the river.

But the river will only help them hide if you guys "catch" all the fish.

Now you will work in pairs. I give each pair an envelope - a net in which three fish are entangled. Your task is to get all the fish, write down number 1 and solve

Fish tasks. Prove that the numbers are coprime

1) 40 and 15 2) 45 and 49 3) 16 and 21

Mutual verification. Pay attention to the evaluation criteria. Slide 6-7

Generalization: How to prove that numbers are coprime?

Rated.

Well done. Helped a girl and a boy. The river covered them under its bank. Geese-swans flew by.

As a sign of gratitude, the Boy will spend a physical minute for you (video) Slide 9

In which case will the apple tree hide them?

If a girl tries her forest apple.

Right. Let's all "eat" forest apples together. And the apples on it are not simple, with unusual tasks, called LOTTO. We “eat” large apples one per group, i.e. we work in groups. Find the GCD in each cell on the small answer cards. When all the cells are closed, turn the cards over and you should get a picture.

Tasks on forest apples

Find GCD:

1 group		2 group
gcd(48,84)=	GCD (60.48)=	gcd(60,80)=	GCD (80.64)=
gcd (12,15)=	gcd(15,20)=	GCD (50.30)=	gcd (12,16)=
3 group		4 group
GCD (123.72)=	gcd(120,96)=	gcd(90,72)=	GCD(15;100)=
gcd(45,30)=	GCD (15.9)=	gcd(14,42)=	GCD (34.51)=

Check: I go through the rows, check the picture

Generalization: What needs to be done to find the GCD?

Well done. The apple tree covered them with branches, covered them with leaves. Geese - swans lost them and flew on. What next?

They ran again. It was not far away, then the geese saw them, began to beat their wings, they want to snatch their brother out of their hands. They ran to the stove. The stove will hide them if the girl tries the rye pie.

Let's help the girl.Assignment by options, test

TEST

Subject

Option 1

1. Which numbers are common divisors of 24 and 16?

1) 4, 8; 2) 6, 2, 4;

3) 2, 4, 8; 4) 8, 6.

1. Is 9 the greatest common divisor of 27 and 36?

1. Yes; 2) no.

1. Given the numbers 128, 64 and 32. Which one is the greatest divisor of all three numbers?

1) 128; 2) 64; 3) 32.

1. Are the numbers 7 and 418 coprime?

1) yes; 2) no.

1) 5 and 25;

2) 64 and 2;

3) 12 and 10;

4) 100 and 9.

TEST

Subject : NOD. Coprime numbers.

Option 1

1. Which numbers are common divisors of 18 and 12?

1) 9, 6, 3; 2) 2, 3, 4, 6;

3) 2, 3; 4) 2, 3, 6.

1. Is 4 the greatest common divisor of 16 and 32?

1. Yes; 2) no.

1. Given the numbers 300, 150 and 600. Which one is the greatest divisor of all three numbers?

1) 600; 2) 150; 3) 300.

1. Are the numbers 31 and 44 coprime?

1) yes; 2) no.

1. Which of the numbers are relatively prime?

1) 9 and 18;

2) 105 and 65;

3) 44 and 45;

4) 6 and 16.

Examination. Self-check from a slide. Evaluation criteria. Slide 10-11

Well done. They ate pies. The girl and her brother sat in the stoma and hid. Geese-swans flew-flew, shouted-shouted and flew away to Baba Yaga with nothing.

The girl thanked the stove and ran home.

Soon both father and mother came home from work.

Summary of the lesson. While we were helping a girl with a boy, what topics did we repeat? (Finding the gcd of two numbers, coprime numbers.)

How to find the GCD of several natural numbers?

How to prove that numbers are coprime?

During the lesson, for each task, I gave you grades and you evaluated yourself. By comparing them, the average score for the lesson will be set.

Reflection.

Dear friends! Summing up the lesson, I would like to hear your opinion about the lesson.

• What was interesting and instructive in the lesson?
• Can I be sure that you can handle this type of task?
• Which of the tasks turned out to be the most difficult?
• What knowledge gaps emerged in the lesson?
• What problems did this lesson give rise to?
• How do you assess the role of the teacher? Did it help you acquire the skills and knowledge to solve these types of problems?

Glue the apples to the tree. Who coped with all the tasks, and everything was clear - glue a red apple. Who had a question - green, who did not understand - yellow. slide 12

Is the statement true? The smallest two-digit prime number is 11

Is the statement true? The largest two-digit composite number is 99

Is the statement true? The numbers 8 and 10 are coprime

Is the statement true? Some composite numbers cannot be factored into prime factors

Key to the dictation: _ /\ _ _ /\ /\ Evaluation criteria No errors - "5" 1-2 errors - "4" 3 errors - "3" More than three - "2"

Prove that the numbers 16 and 21 are relatively prime 3 Prove that the numbers 40 and 15 are relatively prime Prove that the numbers 45 and 49 are relatively prime 2 1 40=2 2 2 5 15=3 5 gcd(40; 15) =5, non-prime numbers 45=3 3 5 49=7 7 gcd(45; 49)=, coprime numbers 16=2 2 2 2 21=3 7 gcd(45; 49) =1, coprime numbers

Evaluation criteria No errors - "5" 1 error - "4" 2 errors - "3" More than two - "2"

Group 1 GCD(48.84)= GCD(60.48)= GCD(12.15)= GCD(15.20)= Group 3 GCD(123.72)= GCD(120.96)= GCD(45, 30)= GCD(15.9)= Group 2 GCD(60.80)= GCD(80.64)= GCD(50.30)= GCD(12.16)= Group 4 GCD(90.72)= GCD (15.100)= GCD (14.42)= GCD(34.51)=

Tasks from the stove B1 3 2. 1 3. 3 4. 1 5. 4 B2 4 2. 2 3. 2 4. 1 5. 3

Evaluation criteria No errors - "5" 1-2 errors - "4" 3 errors - "3" More than three - "2"

Reflection I understood everything, I coped with all the tasks, there were minor difficulties, but I coped with them, there were a few questions left