Greatest common divisor of coprime numbers. To the task on the board record

25.01.2021

Remember!

If a natural number is only divisible by 1 and itself, then it is called prime.

Any natural number is always divisible by 1 and itself.

The number 2 is the smallest prime number. This is the only even prime number, the rest prime numbers- odd.

There are many prime numbers, and the first among them is the number 2. However, there is no last prime number. In the "For Study" section, you can download a table of prime numbers up to 997.

But many natural numbers are evenly divisible by other natural numbers.

For example:

the number 12 is divisible by 1, by 2, by 3, by 4, by 6, by 12;
36 is divisible by 1, by 2, by 3, by 4, by 6, by 12, by 18, by 36.

The numbers by which the number is evenly divisible (for 12 these are 1, 2, 3, 4, 6 and 12) are called the divisors of the number.

Remember!

The divisor of a natural number a is such a natural number that divides the given number "a" without a remainder.

A natural number that has more than two factors is called a composite number.

Note that the numbers 12 and 36 have common divisors. These are numbers: 1, 2, 3, 4, 6, 12. The largest divisor of these numbers is 12.

The common divisor of two given numbers "a" and "b" is the number by which both given numbers "a" and "b" are divided without a remainder.

Remember!

Greatest Common Divisor(GCD) of two given numbers "a" and "b" is largest number, by which both numbers "a" and "b" are divisible without a remainder.

Briefly, the greatest common divisor of the numbers "a" and "b" is written as follows:

gcd (a; b) .

Example: gcd (12; 36) = 12 .

The divisors of numbers in the notation of the solution denote capital letter"D".

D(7) = (1, 7)

D(9) = (1, 9)

gcd (7; 9) = 1

The numbers 7 and 9 have only one common divisor - the number 1. Such numbers are called coprime numbers.

Remember!

Coprime numbers are natural numbers that have only one common divisor - the number 1. Their GCD is 1.

How to find the greatest common divisor

To find the gcd of two or more natural numbers you need:

decompose the divisors of numbers into prime factors;

Calculations are conveniently written using a vertical bar. To the left of the line, first write down the dividend, to the right - the divisor. Further in the left column we write down the values of private.

Let's explain right away with an example. Let's factorize the numbers 28 and 64 into prime factors.

Underline the same prime factors in both numbers.
28 = 2 2 7
64 = 2 2 2 2 2 2
We find the product of identical prime factors and write down the answer;
GCD (28; 64) = 2 2 = 4
Answer: GCD (28; 64) = 4

You can arrange the location of the GCD in two ways: in a column (as was done above) or “in a line”.

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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 is an even number, therefore, it is divisible by 2, then 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. By successive division is meant the division of a larger number by a smaller one, a smaller number by the first remainder, the first remainder by 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?

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.

Solution:

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.)

Solution:

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.

Solution:

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 everything double figures, which have the same property. (113, 131; 13, 31; 17, 71; 37, 73; 79, 97.)

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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:

Promote development logical thinking students, speech and mental operations skills;
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:

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.

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 factored into prime factors; (not true).

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

Evaluated their oral work in the evaluation sheet.

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: they rushed in the distance Swan geese 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. So?

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

Topic

Option 1

Which numbers are common divisors of 24 and 16?

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

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

Is 9 the greatest common divisor of 27 and 36?

Yes; 2) no.

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

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

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

Topic : NOD. Coprime numbers.

Option 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.

Is 4 the greatest common divisor of 16 and 32?

Yes; 2) no.

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

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

Are the numbers 31 and 44 coprime?

1) yes; 2) no.

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. Comparing them will be exhibited GPA for the lesson.

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

Solving problems from the problem book Vilenkin, Zhokhov, Chesnokov, Schwarzburd for grade 6 in mathematics on the topic:

Chapter I. Ordinary fractions.
§ 1. Divisibility of numbers:
6. Greatest common divisor. Coprime numbers

146 Find all the common divisors of the numbers 18 and 60; 72, 96 and 120; 35 and 88.
SOLUTION

147 Find the prime factorization of the greatest common divisor of a and b if a = 2 2 3 3 and b = 2 3 3 5; a = 5 5 7 7 7 and b = 3 5 7 7.
SOLUTION

148 Find the greatest common divisor of the numbers 12 and 18; 50 and 175; 675 and 825; 7920 and 594; 324, 111 and 432; 320, 640 and 960.
SOLUTION

149 Are the numbers 35 and 40 coprime; 77 and 20; 10, 30, 41; 231 and 280?
SOLUTION

150 Are the numbers 35 and 40 coprime; 77 and 20; 10, 30, 41; 231 and 280?
SOLUTION

151 Write down all proper fractions with a denominator of 12 whose numerator and denominator are relatively prime numbers.
SOLUTION

152 The guys received the same gifts on the New Year tree. All gifts together contained 123 oranges and 82 apples. How many children were present at the Christmas tree? How many oranges and how many apples were in each gift?
SOLUTION

153 For a trip outside the city, several buses were allocated to the plant's employees, with the same number of seats. 424 people went to the forest, and 477 went to the lake. All seats on the buses were occupied, and not a single person was left without a seat. How many buses were allocated and how many passengers were on each of them?
SOLUTION

154 Calculate verbally in a column
SOLUTION

155 Using Figure 7, determine if the numbers a, b, and c are prime.
SOLUTION

156 Is there a cube whose edge is expressed natural number and for which the sum of the lengths of all edges is expressed as a prime number; surface area expressed as a prime number?
SOLUTION

157 Factorize the numbers 875; 2376; 5625; 2025; 3969; 13125.
SOLUTION

158 Why, if one number can be decomposed into two prime factors, and the second - into three, then these numbers are not equal?
SOLUTION

159 Is it possible to find four distinct prime numbers such that the product of two of them is equal to the product of the other two?
SOLUTION

160 In how many ways can 9 passengers be accommodated in a nine-seater minibus? In how many ways can they accommodate themselves if one of them, who knows the route well, sits next to the driver?
SOLUTION

161 Find the values of expressions (3 8 5-11):(8 11); (2 2 3 5 7):(2 3 7); (2 3 7 1 3):(3 7); (3 5 11 17 23):(3 11 17).
SOLUTION

162 Compare 3/7 and 5/7; 11/13 and 8/13;1 2/3 and 5/3; 2 2/7 and 3 1/5.
SOLUTION

163 Use a protractor to plot AOB=35° and DEF=140°.
SOLUTION

164 1) Beam OM divided the developed angle AOB into two: AOM and MOB. The AOM angle is 3 times the MOB. What are the angles AOM and BOM. Build them. 2) Beam OK divided the developed angle COD into two: SOK and KOD. The SOC angle is 4 times less than KOD. What are the angles COK and KOD? Build them.
SOLUTION

165 1) Workers repaired an 820 m long road in three days. On Tuesday they repaired 2/5 of this road, and on Wednesday 2/3 of the rest. How many meters of the road did the workers repair on Thursday? 2) The farm contains cows, sheep and goats, a total of 3400 animals. Sheep and goats together make up 9/17 of all animals, and goats make up 2/9 of the total number of sheep and goats. How many cows, sheep and goats are on the farm?
SOLUTION

166 Present as common fraction numbers 0.3; 0.13; 0.2 and in the form decimal fraction 3/8; 4 1/2; 3 7/25
SOLUTION

167 Perform the action, writing each number as a decimal fraction 1/2 + 2/5; 1 1/4 + 2 3/25
SOLUTION

168 Express as the sum of prime terms the numbers 10, 36, 54, 15, 27 and 49 so that there are as few terms as possible. What suggestions can you make about representing numbers as a sum of prime terms?
SOLUTION

169 Find the greatest common divisor of a and b if a = 3 3 5 5 5 7, b = 3 5 5 11; a = 2 2 2 3 5 7, b = 3 11 13 .