Competition for young teachers

Bryansk region

"Pedagogical Debut - 2014"

2014-2015 academic year

Anchorage lesson in mathematics in grade 6

on the topic “GCD. Mutually prime numbers "

Work place:MBOU "Glinischevskaya secondary school" of the Bryansk region

Goals:

Educational:

• To consolidate and organize the studied material;
• Practice the skills of decomposing numbers into prime factors and finding GCD;
• Test students' knowledge and identify gaps;

Developing:

• Promote the development of students' logical thinking, speech and mental operations skills;
• Promote the formation of the ability to notice patterns;
• Promote an increase in the level of mathematical culture;

Educational:

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

Lesson type: a lesson in generalization and systematization of knowledge.

Teaching methods : explanatory and illustrative, independent work.

Equipment: computer, screen, presentation, handouts.

During the classes:

1. Organizing time.

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

You sat down quietly 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 "Scorecard", i.e. in addition to my assessment, you will assess yourself by completing each task.

Evaluation paper

Guys, what topic did you study over several lessons? (Learned to find the greatest common factor).

What do you think we will do with you today? Formulate the topic of our lesson. (Today we will continue to work with the greatest common divisor. The topic of our lesson: “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 lesson topic: Greatest Common Divisor. Mutually prime numbers ”.

1. Knowledge update

Several theoretical questions

Is the statement correct. "Yes" - __; "No" - /\. Slide 3-4

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

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

Assessed their oral work on the score sheet.

1. Systematization of knowledge

There will be some magic in our lesson today.

Where does magic meet? (in a fairy tale)

Guess from the drawing which fairy tale we will find ourselves in. ( Slide 5 ) The Tale of the Geese-Swans. Absolutely right. Well done. 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 their brother.

She put my brother on the grass under the window, and she ran outside, played, took a walk. When the girl returned, the brother was gone. She started looking for him, she shouted, called him, but no one responded. She ran out into an open field and only saw: the geese darted in the distance and disappeared behind a dark forest. Then the girl realized that they had taken her brother away. She had known for a long time that the goose-swans carried away small children.

She rushed after them. On the way, she met a stove, an apple tree, a river. But our river is not dairy in the jelly banks, but the usual one, in which there are a lot of 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. Day is already leaning towards evening, suddenly she sees - there is a hut on a chicken leg, with one window, turning around itself. In the hut, 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 not for the little mouse, which she fed with porridge, Baba Yaga would have fried it in the oven and ate it. The girl quickly grabbed her brother and ran home. Geese - swans noticed them and flew in pursuit. And whether they get home safely - everything now depends on us guys. Let's continue the story.

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

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

You will now 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 No. 1 and solve

Quests for the fish. Prove that the numbers are coprime

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

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

Generalization: How to prove that numbers are coprime?

Have given an assessment.

Well done. Helped the girl with the boy. The river covered them under its bank. The swan geese flew past.

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

In what case will the apple tree hide them?

If a girl tastes her forest apple.

Right. Let us all "eat" forest apples together. And the apples on it are not simple, with unusual tasks, called LOTO. Big apples "eat" one per group, ie. we work in groups. Find the GCD in each box on the small answer cards. When all the cells are closed, turn the cards over and you should get a picture.

Forest apples quests

Find the GCD:

1st group		Group 2
GCD (48.84) =	GCD (60.48) =	GCD (60,80) =	GCD (80.64) =
GCD (12.15) =	GCD (15.20) =	GCD (50.30) =	GCD (12.16) =
Group 3		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 checking the picture

Summary: What do you need to do 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 what is next?

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

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

TEST

Theme

Option 1

1. Which numbers are common factors for 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. The given numbers are 128, 64 and 32. Which of them is the greatest divisor of all three numbers?

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

1. Are the numbers 7 and 418 mutually prime?

1) yes; 2) no.

1) 5 and 25;

2) 64 and 2;

3) 12 and 10;

4) 100 and 9.

TEST

Theme : GCD. Mutually prime numbers.

Option 1

1. Which numbers are common factors for 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 numbers 300, 150 and 600. Which of them is the greatest divisor of all three numbers?

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

1. Are the numbers 31 and 44 mutually prime?

1) yes; 2) no.

1. Which numbers are coprime?

1) 9 and 18;

2) 105 and 65;

3) 44 and 45;

4) 6 and 16.

Examination. Self-test from the slide. Evaluation criteria. Slide 10-11

Well done. We ate the pies. The girl and her brother sat down in the stomata 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 my father and mother came home from work.

Lesson summary. While we were helping the girl and the 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 the numbers are coprime?

During the lesson, for each assignment, I gave you grades and you graded yourself. By comparing them, the average grade 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 will cope with this type of task?
• Which of the tasks turned out to be the most difficult?
• What gaps in knowledge were revealed in the lesson?
• What problems did this lesson generate?
• How do you assess the role of the teacher? Has he helped you acquire the skills and knowledge to solve this type of problem?

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

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

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

Is the statement true? Numbers 8 and 10 are relatively prime

Is the statement true? Some composite numbers cannot be factorized

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

Prove that 16 and 21 are coprime 3 Prove that 40 and 15 are coprime Prove that 45 and 49 are coprime 2 1 40 = 2 2 2 5 15 = 3 5 GCD (40; 15) = 5, numbers are not coprime 45 = 3 3 5 49 = 7 7 GCD (45; 49) =, numbers are coprime 16 = 2 2 2 2 21 = 3 7 GCD (45; 49) = 1, the numbers are coprime

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 some difficulties, but I coped with them there were a few questions

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

Solution. Each of the numbers 48 and 36 must be divisible by the number of gifts. Therefore, first we 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.

Common divisors of 48 and 36 are 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.

This means that 12 gifts can be made. Each gift will contain 4 Swallow sweets (48: 12 = 4) and 3 Cheburashka sweets (36: 12 = 3).

Lesson content lesson outline support frame lesson presentation accelerative methods interactive technologies Practice tasks and exercises self-test workshops, trainings, cases, quests homework discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photos, pictures, charts, tables, schemes humor, anecdotes, fun, comics parables, sayings, crosswords, quotes Add-ons abstracts articles chips for the curious cheat sheets textbooks basic and additional vocabulary of terms others Improving textbooks and lessonsbug fixes in the tutorial updating a fragment in the textbook elements of innovation in the lesson replacing outdated knowledge with new ones For teachers only perfect lessons calendar plan for the year methodological recommendations of the discussion program Integrated lessons

In this article, we will talk about what coprime numbers are. In the first section, we formulate definitions for two, three or more coprime numbers, give several examples and show in which cases two numbers can be considered prime with respect to each other. After that, let's move on to the formulation of the main properties and their proofs. In the last paragraph, we will talk about a related concept - pairwise primes.

What are coprime numbers

Two or more integers can be mutually prime. To begin with, we introduce a definition for two numbers, for which we need the concept of their greatest common divisor. If necessary, repeat the material dedicated to him.

Definition 1

Two such numbers a and b will be mutually prime, the greatest common divisor of which is 1, i.e. GCD (a, b) = 1.

From this definition, we can conclude that the only positive common divisor of two coprime numbers will be equal to 1. Only two such numbers have two common factors - one and minus one.

What are some examples of mutually prime numbers? For example, such a pair would be 5 and 11. They have only one common positive divisor equal to 1, which is a confirmation of their mutual simplicity.

If we take two primes, then in relation to each other they will be mutually prime in all cases, but such mutual relations are also formed between composite numbers. There are cases when one number in a pair of mutually prime is composite, and the second is prime, or both of them are composite.

This statement is illustrated by the following example: composite numbers - 9 and 8 form a coprime pair. Let us prove this by calculating their greatest common divisor. To do this, write down all their divisors (we recommend re-reading the article on finding the divisors of a number). For 8, these will be the numbers ± 1, ± 2, ± 4, ± 8, and for 9 - ± 1, ± 3, ± 9. We choose from all the divisors the one that will be common and the largest - this is one. Therefore, if GCD (8, - 9) = 1, then 8 and - 9 will be mutually prime with respect to each other.

500 and 45 are not mutually prime numbers, because they have another common divisor - 5 (see the article on criteria for divisibility by 5). Five is greater than one and is a positive number. Another similar pair can be - 201 and 3, since both of them can be divided by 3, as indicated by the corresponding divisibility criterion.

In practice, it is quite often necessary to determine the mutual simplicity of two integers. Finding out this can be reduced to finding the greatest common divisor and comparing it with unity. It is also convenient to use the table of prime numbers so as not to make unnecessary calculations: if one of the given numbers is in this table, then it is divisible only by one and by itself. Let's analyze the solution to a similar problem.

Example 1

Condition: find out if 275 and 84 are coprime.

Solution

Both numbers clearly have more than one divisor, so we cannot immediately call them coprime.

Calculate the greatest common divisor using Euclid's algorithm: 275 = 84 3 + 23, 84 = 23 3 + 15, 23 = 15 1 + 8, 15 = 8 1 + 7, 8 = 7 1 + 1, 7 = 7 1.

Answer: since gcd (84, 275) = 1, then these numbers will be relatively prime.

As we said earlier, the definition of such numbers can be extended to cases when we have not two numbers, but more.

Definition 2

Integers a 1, a 2,…, a k, k> 2 will be mutually prime if they have the greatest common divisor equal to 1.

In other words, if we have a set of some numbers with the largest positive divisor greater than 1, then all these numbers are not mutually inverse with respect to each other.

Let's take a few examples. So, the integers - 99, 17 and - 27 - are coprime. Any number of primes will be mutually prime to all members of the population, such as in the sequence 2, 3, 11, 19, 151, 293, and 667. But the numbers 12, - 9, 900 and − 72 They will not be coprime, because apart from unity they will have one more positive divisor equal to 3. The same applies to the numbers 17, 85 and 187: apart from one, they can all be divided by 17.

Usually, the mutual simplicity of numbers is not obvious at first glance, this fact needs to be proved. To find out whether some numbers are relatively prime, you need to find their greatest common divisor and draw a conclusion based on its comparison with unity.

Example 2

Condition: Determine if the numbers 331, 463, and 733 are coprime.

Solution

Check with the table of prime numbers and determine that all three of these numbers are in it. Then only one can be their common divisor.

Answer: all these numbers will be mutually prime with respect to each other.

Example 3

Condition: provide proof that the numbers - 14, 105, - 2 107 and - 91 are not coprime.

Solution

Let's start by identifying their greatest common divisor, and then make sure that it is not equal to 1. Since negative numbers have the same divisors as the corresponding positive ones, then GCD (- 14, 105, 2 107, - 91) = GCD (14, 105, 2 107, 91). According to the rules that we gave in the article on finding the greatest common divisor, in this case the GCD will be equal to seven.

Answer: seven is more than one, which means that these numbers are not mutually prime.

Basic properties of coprime numbers

Such numbers have some practically important properties. Let us list them in order and prove.

Definition 3

If we divide the integers a and b by the number corresponding to their greatest common divisor, we get coprime numbers. In other words, a: gcd (a, b) and b: gcd (a, b) will be relatively prime.

We have already proved this property. The proof can be found in the article on the properties of the greatest common divisor. Thanks to him, we can determine pairs of mutually prime numbers: just take any two integers and divide by GCD. As a result, we should get mutually prime numbers.

Definition 4

A necessary and sufficient condition for the mutual simplicity of the numbers a and b is the existence of such integers u 0 and v 0 for which the equality a u 0 + b v 0 = 1 will be true.

Proof 1

Let's start by proving the necessity of this condition. Let's say we have two coprime numbers, denoted a and b. Then, by the definition of this concept, their greatest common divisor will be equal to one. From the properties of GCD, we know that for integers a and b there is a Bezout relation a u 0 + b v 0 = gcd (a, b)... From it we get that a u 0 + b v 0 = 1... After that, we need to prove the sufficiency of the condition. Let the equality a u 0 + b v 0 = 1 will be true, in that case, if Gcd (a, b) divides and a , and b, then it will divide and the sum a u 0 + b v 0, and unity, respectively (this can be asserted from the divisibility properties). And this is possible only if Gcd (a, b) = 1, which proves the mutual simplicity of a and b.

Indeed, if a and b are coprime, then according to the previous property, the equality a u 0 + b v 0 = 1... We multiply both sides by c and get that a c u 0 + b c v 0 = c... We can split the first term a c u 0 + b c v 0 by b, because this is possible for a · c, and the second term is also divisible by b, because one of the factors we have is b. From this we conclude that the entire amount can be divided by b, and since this sum is equal to c, then c can be divided by b.

Definition 5

If two integers a and b are coprime, then GCD (a c, b) = GCD (c, b).

Proof 2

Let us prove that GCD (a c, b) will divide GCD (c, b), and after that, that GCD (c, b) divides GCD (a c, b), which will prove that the equality GCD (a C, b) = gcd (c, b).

Since gcd (ac, b) divides both ac and b, and gcd (ac, b) divides b, it will also divide b c. Hence, GCD (a c, b) divides both ac and b c, therefore, by virtue of the properties of GCD, it also divides GCD (ac, b c), which will be equal to c GCD (a, b ) = c. Therefore, GCD (a c, b) divides both b and c, therefore, GCD (c, b) also divides.

You can also say that since GCD (c, b) divides both c and b, it will divide both c and a · c. Hence, GCD (c, b) divides both ac and b, therefore, GCD (ac, b) also divides.

Thus, gcd (a c, b) and gcd (c, b) mutually share each other, which means that they are equal.

Definition 6

If the numbers from the sequence a 1, a 2,…, a k will be coprime with respect to the numbers of the sequence b 1, b 2, ..., b m(for natural values ​​of k and m), then their products a 1 · a 2 ·… · a k and b 1 b 2… b m are also coprime, in particular, a 1 = a 2 =… = a k = a and b 1 = b 2 =… = b m = b, then a k and b m- mutually simple.

Proof 3

According to the previous property, we can write the equalities of the following form: GCD (a 1 · a 2 ·… · a k, b m) = GCD (a 2 ·… · a k, b m) =… = GCD (a k, b m) = 1. The possibility of the last transition is ensured by the fact that a k and b m are mutually simple by condition. Hence, GCD (a 1 · a 2 ·… · a k, b m) = 1.

We denote a 1 a 2 ... ak = A and we obtain that GCD (b 1 b 2 ... bm, a 1 a 2 ... ak) = GCD (b 1 b 2 ... bm , A) = GCD (b 2 ... b bm, A) = ... = GCD (bm, A) = 1. This will be true due to the last equality from the chain constructed above. Thus, we have obtained the equality GCD (b 1 b 2… b m, a 1 a 2… a k) = 1, which can be used to prove the mutual simplicity of the products a 1 · a 2 ·… · a k and b 1 b 2… b m

These are all the properties of coprime numbers that we would like to tell you about.

The concept of pairwise primes

Knowing what coprime numbers are, we can formulate a definition of pairwise primes.

Definition 7

Pairwise primes Is a sequence of integers a 1, a 2,…, a k, where each number will be mutually prime with respect to the others.

An example of a sequence of pairwise primes would be 14, 9, 17, and - 25. Here all pairs (14 and 9, 14 and 17, 14 and - 25, 9 and 17, 9 and - 25, 17 and - 25) are coprime. Note that the condition of mutual simplicity is mandatory for pairwise primes, but coprime numbers will not be pairwise prime in all cases. For example, in the sequence 8, 16, 5, and 15, the numbers are not, since 8 and 16 will not be coprime.

You should also dwell on the concept of a collection of a certain number of primes. They will always be both mutually and pairwise simple. An example would be sequence 71, 443, 857, 991. In the case of prime numbers, the concepts of mutual and pairwise simplicity will coincide.

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Common divisors

Example 1

Find the common divisors of $ 15 $ and $ –25 $.

Solution.

Divisors of the number $ 15: $ 1, 3, 5, 15 and their opposite.

Divisors of the number $ –25: $ 1, 5, 25 and their opposite.

Answer: the numbers $ 15 $ and $ –25 $ have common divisors of $ 1, $ 5 and their opposite.

According to the divisibility properties of the number $ −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 –а will also be a common divisor for these numbers.

Most often, in practice, they are limited to only positive divisors, but do not forget that each integer opposite to a positive divisor will also be a divisor of this number.

Determining the Greatest Common Divisor (GCD)

According to the properties of divisibility, every integer has at least one nonzero divisor, and the number of such divisors is finite. In this case, the common divisors of the given numbers are also finite. Of all the common divisors of the given numbers, the largest number can be selected.

If all these numbers are equal to zero, it is impossible to determine the greatest of the common divisors, since zero is divisible by any integer, of which there is an infinite number.

The greatest common divisor of the numbers $ a $ and $ b $ in mathematics is denoted $ gcd (a, b) $.

Example 2

Find the gcd of integers $ 412 and $ –30 $ ..

Solution.

Let's find the divisors of each of the numbers:

$ 12 $: numbers $ 1, 3, 4, 6, 12 $ and their opposite.

$ –30 $: numbers $ 1, 2, 3, 5, 6, 10, 15, 30 $ and their opposite.

Common divisors of $ 12 $ and $ –30 $ are $ 1, 3, 6 $ and their opposite.

$ Gcd (12, –30) = 6 $.

Determining the GCD of three or more integers can be similar to the definition of the GCD of two numbers.

GCD of three or more integers is the largest integer that divides all numbers at the same time.

Designate the greatest 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 the divisors of each of the numbers:

$ –12 $: numbers $ 1, 2, 3, 4, 6, 12 $ and their opposite;

$ 32: numbers $ 1, 2, 4, 8, 16, 32 $ and their opposite;

$ 56: The numbers $ 1, 2, 4, 7, 8, 14, 28, 56 $ and their opposite.

Common divisors of $ –12, 32, 56 $ are $ 1, 2, 4 $ and their opposite.

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 may be one of these numbers.

Mutually prime numbers

Definition 3

Integers $ a $ and $ b $ - mutually simple if $ gcd (a, b) = 1 $.

Example 4

Show that the numbers $ 7 $ and $ 13 $ are coprime.