Solving problems from the problem book Vilenkin, Zhokhov, Chesnokov, Schwarzburd for grade 6 in mathematics on the topic:
§ 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 it all down 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 Use Figure 7 to determine if the numbers a, b, and c are prime.
SOLUTION
156 Is there a cube whose edge is expressed by a natural number and for which the sum of the lengths of all edges is expressed by 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 .
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 .
Solution . Let's decompose these numbers into prime factors:
The prime factor 2 is included in the first factorization to the power of 2, in the second factorization to the power of 1, and to 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:
- 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 notebooks, write down the number, class work 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: 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. So what is 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
- 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 largest divisor 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
Subject : 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
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 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 a year guidelines discussion programs 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 min.)
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 Rus' 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 the more, by which each of the given natural numbers is divisible). 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) arrange clarification general new knowledge (the ability to apply a new method of action to solve all tasks of a given type).
Organization 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.