How to know the smallest possible multiple of numbers. The value of the least common multiple: how, apply the value of the NOC The value of the lowest common multiple of negative numbers

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In order to understand how to calculate the NOC, the following should be distinguished from the meanings of the term "multiple".

A multiple of A is such a natural number that it can be divided by A without too much. So, multiples of 5 can be 15, 20, 25 and so on.

The number of dilniks of a specific number can be marked by a quantity, and the axis of multiples is impersonal.

A real multiple of natural numbers is a number that can be divided by them without excess.

How to know the least common multiple of numbers

The least global multiple (LCM) of numbers (two, three, or more) is the least natural number, as it can be divided by all numbers of numbers.

To know the NOC, you can pick up a sprinkling of methods.

For small numbers, you can manually write down a row of all multiples of these numbers doti, and there are no commonplaces in the middle. Multiple signify in the record with the great letter Do.

For example, a multiple of 4 can be written like this:

Up to (4) = (8,12, 16, 20, 24, ...)

Before (6) = (12, 18, 24, ...)

So, you can tell that the smallest common multiple of the numbers 4 and 6 is the number 24. This record should be written in this order:

LCM(4, 6) = 24

Even though the numbers are great, to know a total multiple of three or more numbers, it is better to win another way to calculate the NOC.

For vykonannya zavdannya it is necessary to lay out the proponated numbers into simple multipliers.

It is necessary to write on the back of the head in a row of the largest of the numbers, and under it - reshtu.

At the location of the skin number, there may be a difference in the number of multiples.

For example, let's put the numbers 50 and 20 into simple numbers.

In the distribution of the smallest number, there are next multipliers, as in the distribution of the first largest number, and then add to the new one. The pointed butt does not have doubles.

Now you can virahuvati in the least severe multiple of 20 and 50.

LCM (20, 50) = 2 * 5 * 5 * 2 = 100

So, the addition of prime multiples of a larger number and multiples of another number, which did not reach the distribution of a larger one, will be the smallest common multiple.

To know the NOC of three numbers and more, the next ones are laid out on simple multipliers, like i in the forward fall.

As a butt, you can know the smallest multiple of the numbers 16, 24, 36.

36 = 2 * 2 * 3 * 3

24 = 2 * 2 * 2 * 3

16 = 2 * 2 * 2 * 2

So, at the layout of a larger number, only two doubles from the layout of sixteen (one є in the layout of twenty four or three) did not increase into multipliers.

In this rank, it is necessary to add them to the laying out of a larger number.

LCM (12, 16, 36) = 2 * 2 * 3 * 3 * 2 * 2 = 9

Іsnuyut okremі vpadki vyznachennya least zagalnogo multiple. So, if one of the numbers can be added without too much to another, then more of these numbers will be the smallest global multiple.

For example, NOK twelve that twenty chotirioh will be twenty chotiri.

It is necessary to know the least significant multiple of mutually prime numbers, so that you can find the same dilniks, their NOCs are dearer to their creation.

For example, LCM (10, 11) = 110.

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Captions before slides:

Math lesson for grade 6. Teacher of mathematics SBOU ZOSh No. 539 Dmitro Vadimovich Labzіn. The smallest multiple.

Sleepy robot. 1. Calculate: a)? ? 2. Vіdomo, scho guess the right way, vikoristovuyuchi terms: "є dilnik", "divided", "є multiple". What are synonyms? 3. You can confirm that the numbers a, b and c are multiples of 14, so: - Know privately how the number a is divided by 14, the number b by 14.

Letter. 2. To find a splinter of wild multiple numbers 15 and 30. Solution. Multiples of 15: 15; thirty; 45; 60; 75; 90… Multiples of 30: 30; 60; 90… Large multiples: 30; 60; 90. - Name the smallest major multiple of the numbers 15 and 30. - The number 30. - Try to formulate how to call a number the smallest major multiple of two natural numbers a and b? The smallest global multiple of natural numbers a and b is the smallest natural number, i.e. a multiple of і a, і b. - Tell me, be kind, what is the best way to know the NOK? - Why? LCM(15; 30) = 30. Write:

2. Given numbers: - Think, how can you know the smallest multiple of the numbers a and b? Algorithm 1. Expand given numbers into prime factors; 2. Write down one of them; 3. Add daily multipliers from the spread of the next number; 4. Know the negative tvir.

Butt 1. Know the NOC (32; 25). Solution. Let's decompose the numbers 32 and 25 into simple multipliers. ; - What can you say about the numbers 32 and 25? The smallest mutually multiple of prime numbers is good for their creation. Butt 2. Know the LCM of numbers 12; 15; twenty; 60. Solution. If the middle of the numbers is the same, which is divided into the reshta, then the tse i є NOK of these numbers. - What did you remember?

Given numbers: 15 and 30. Multiples of 15: 15; thirty; 45; 60; 75; 90… Multiples of 30: 30; 60; 90… The smallest multiple: 30. Tse kavo! Multiples of 30: 30; 60; 90… A skin multiple of the number of LCMs (a; b) is a common multiple of the numbers a and b i, now, a skin multiple of a multiple of the numbers of LCMs (a; b).

Topic: “The smallest number of times”, Grade 6, EMC Vilenkin N.Ya.

lesson type: "revealing" new knowledge

Golovnі tsіlі.

Induce the designation of the smallest common multiple of the LCM value algorithm. Form the ability to know the NOC.

Train building

Before trying to understand a simple warehouse number;

Identity sign for 2, 3, 5, 9, 10:

Other ways to know the NOC:

Algorithm znakhodzhennya peretina that o'dnannya mnozhin;

3) Train building layout on simple multipliers.

I Self-determination to the extent of activity.

Let's do a workout. Children are divided into groups for options. The first to take a card from the leaders and to stun your group:

1st - sign of fakeness for 2;

The other is a sign of authenticity by 3;

the third - a sign of fakeness by 5;

The fourth - a sign of authenticity on 9;

5th - sign of fakeness by 10;

6th - a sign of fakeness for 2..

The numbers appear on the presentation screen: 51, 22, 37, 191, 163, 88, 47, 133, 152, 202, 403, 75, 507, 609, 708 (otherwise they are lifted from the month, so that until the number you can stop the sign given to it)

Guys, do you need to know the signs of fakeness? (For spreading numbers into multipliers)

II. Actualization of knowledge

On which class can you beat all the natural numbers for the number of dilniks? (for free and warehouse 1)

What numbers are called simple? (numbers that can be less than two dilniks)

Recycle the sprat of prime numbers) (2,3,5,7,9,11,13,17,…)

Tell me, and for the accomplishment of such tasks, we need to sort out on the basis of variable multipliers? (significance of the biggest infamous dilnik (learned in previous lessons))

What algorithm is the significance of GCD? (The algorithm for the knowledge of the GCD is formulated for the help of spreading into multipliers)

Find the biggest sleeper on 18 and 24?

How do you know. Children are called out in different ways to recognize the GCD (by writing down all the numbers through the spreading on simple multipliers).

Match the GCD with the skin of the numbers.

III. Statement of the initial task and fixation of the aggravated activity

Write down 8 numbers that are multiples of 18 (18, 36, 54, 72, 90, 108. 126, 144)

Write down 6 numbers that are multiples of 24 (24, 48, 72, 96, 120, 144)

Global multiples of these numbers: 72. 144

Give the name of the number 72 (The smallest possible multiple of these numbers: 72)

Otzhe, formulate the topic of today's lesson (at least more than once)

Yaka meta lesson? (learn to know the NOC)

We knew the NOC using the selection method, but what method can you know the NOC? (method of spreading into simple multipliers)

What is the essence of what?

IV. Encourage the project to get out of trouble

At the same time, the algorithm of knowledge of the NOC is formed from the children.

For which requirement:

LCM (18, 24) = 24 * 3 = 72

V. First anchored in the old promotions.

Working zoshit, stor. 28 no. 3 abc

Zavdannya vykonuetsya s komentuvannyam vіdpovіdno to vyvedennym algorithm for zaproponovanoyu scheme.

VI. Self-supporting robot with self-verification for a reason

Learn to win independently No. 181 (abvg)

Spelled right

Pardons are corrected, they appear and are promoted by their reasons.

At this time of learning, yaki correctly scribbled the task, you can dodatkovo robiti No. 183

VII. Inclusion in the system of knowledge and repetition.

Uchnі, yakі pardons were allowed at the independent robotic stage to beat No. 4 of the RT (working zoshit, page 29) for the change of the smallest zagal multiple.

Other studies are violating in groups No. 193, 161, 192

The captains represent the decision.

VIII. Reflection of activity. (The result of the lesson).

- How can a number be called a common multiple of numbers?

How can a number be called the smallest common multiple of numbers?

How to know the least double multiple?

Learn to see the difference between 0 and 1 to place a figurine that depicts the level of understanding of the new topics, for example

IX. Homework.

P.7 stor 29-30, No. 202, 204, 206 (ab) dodatkovo (for the bazhany) No. 209 with a presentation at the next stage.

Let's continue to rozmov about the least double multiple, as we rozpochali at the division "NOC - the least blatant multiple, appointed, apply." In these topics, we can look at ways to find the LCM for three numbers and more, let's take a look at how to know the LCM of a negative number.

Calculation of the lowest common multiple (LCM) through GCD

We have already installed the link of the smallest common multiple from the largest sleeping dilnik. Now we will learn how to sign the NOC through the NOD. Let's pick up the cob, as if it were for positive numbers.

Appointment 1

It is possible to know the smallest number of multiples through the largest possible number using the formula LCM (a, b) = a · b: GCD (a, b).

butt 1

It is necessary to know the LCM of the numbers 126 and 70.

Solution

Let's assume a = 126, b = 70. Let's imagine the value of the formula for calculating the smallest global multiple through the largest half-length LCM (a, b) = a · b: GCD (a, b).

Find the GCD of numbers 70 and 126. For which we need the Euclidean algorithm: 126 \u003d 70 1 + 56, 70 \u003d 56 1 + 14, 56 \u003d 14 4, also, GCD (126 , 70) = 14 .

Let's calculate the NOC: NOK (126, 70) = 126 70: NOD (126, 70) = 126 70: 14 = 630.

Suggestion: LCM (126, 70) = 630 .

butt 2

Find the number 68 and 34.

Solution

GCD is not easy to find in different places, the shards of 68 are divided by 34. Calculate the smallest multiple of the formula: LCM (68, 34) \u003d 68 34: GCD (68, 34) \u003d 68 34: 34 \u003d 68.

Suggestion: NOK (68, 34) = 68.

In this application, the rule of the least common multiple for the number of positive numbers a and b was won: if the first number is divided by another, then the LCM of these numbers is equal to the first number.

Knowledge of the NOC for the help of spreading numbers into simple multipliers

Now let's take a look at the way of knowing the NOC, which is based on the layout of numbers on simple multipliers.

Appointment 2

For the meaning of the smallest common multiple, we need a viconate of low awkward diy:

we add up all the simple multiples of numbers, for which we need to know the LCM;
excluding their otrimanih creations and simple multipliers;
otrimaniy after the inclusion of overhead simple multipliers in tvir dorivnyuє LCM of given numbers.

What is the method of finding the least common multiple bases on equal LCM (a, b) = a · b: GCD (a, b). If you marvel at the formula, then you will understand: the increase in numbers a and b is more expensive in addition to the increase in all multipliers, as if taking part in the distribution of these two numbers. With any GCD of two numbers, it is possible to achieve the completion of all simple multipliers, which are present at the same time in the layouts for the multipliers of these two numbers.

butt 3

We have two numbers 75 and 210. We can factor them out like this: 75 = 3 5 5і 210 = 2 3 5 7. If you add up the sum of all the multiples of the two outgoing numbers, then you will see: 2 3 3 5 5 5 7.

To turn off the capitals for both numbers, the multipliers 3 and 5 are taken out of the offensive form: 2 3 5 5 7 = 1050. This turn will be our NOC for the numbers 75 and 210.

butt 4

Find the LOC of numbers 441 і 700 , having announced insulting numbers into simple multipliers

Solution

We know all the simple multipliers of numbers, given for the mind:

441 147 49 7 1 3 3 7 7

700 350 175 35 7 1 2 2 5 5 7

We take two lances of numbers: 441 \u003d 3 3 7 7 and 700 \u003d 2 2 5 5 7.

Dobutok usikh multipliers, yak took part in the distribution of given numbers, looking at mother: 2 2 3 3 5 5 7 7 7. We know the sleeping multipliers. Tse number 7. Including yoga from the sacred creation: 2 2 3 3 5 5 7 7. Come out, sho nok (441, 700) = 2 2 3 3 5 5 7 7 = 44 100.

Suggestion: LCM (441, 700) = 44100.

There is one more formula for the method of understanding the NOC by the way of spreading numbers into simple multipliers.

Appointment 3

Previously, we included the number of multipliers for both numbers. Now we know it differently:

Let's put offending numbers into simple multipliers:
dodamo to the creation of simple multipliers of the first number of daily multipliers of another number;
otrimaєmo tvir, which will be the noise of the NOC of two numbers.

butt 5

Let's turn to the number 75 and 210, for which we already joked NOC in one of the front butts. Let's put them into simple multipliers: 75 = 3 5 5і 210 = 2 3 5 7. Until the creation of multipliers 3, 5 5 numbers 75 dodamo daily multipliers 2 і 7 numbers 210 . We take: 2 3 5 5 7 . Tse i є NOK numbers 75 and 210.

butt 6

It is necessary to calculate the LCM of the numbers 84 and 648.

Solution

We spread the numbers from the mind into simple multipliers: 84 = 2 2 3 7і 648 = 2 2 2 3 3 3 3. Dodamo to the creation of multipliers 2, 2, 3 7 numbers 84 multipliers 2 , 3 , 3
3 numbers 648 . Take away tvir 2 2 2 3 3 3 3 7 = 4536 . Tse i є is the least significant multiple of the numbers 84 and 648.

Suggestion: LCM (84, 648) = 4536 .

NOC value of three and more numbers

Regardless of how many numbers we can rightly, the algorithm of ours will always be the same: we will sequentially know the LCM of two numbers. On this point, there is a theorem.

Theorem 1

It is acceptable that we may have numbers a 1 , a 2 , … , a k. NOC m k tsikh numbers are rebuffed with sequential calculation m 2 = LCM (a 1, a 2) , m 3 = LCM (m 2 , a 3) , … , m k = LCM (m k − 1 , a k) .

Now let's see how it is possible to put the theorem on the top of specific problems.

butt 7

It is necessary to calculate the smallest possible multiple of four numbers 140, 9, 54 and 250 .

Solution

We introduce the value: a 1 \u003d 140, a 2 \u003d 9, a 3 \u003d 54, a 4 \u003d 250.

Let's just say that m 2 = LCM (a 1, a 2) = LCM (140, 9). We can solve the Euclidean algorithm for calculating the GCD of numbers 140 and 9: 140 = 9 15 + 5, 9 = 5 1 + 4, 5 = 4 1 + 1, 4 = 1 4. It is necessary: GCD (140, 9) = 1, LCM (140, 9) = 140 9: GCD (140, 9) = 140 9: 1 = 1260. Also, m 2 = 1260.

Now, to that algorithm, m 3 = LCM (m 2 , a 3) = LCM (1 260 , 54) . The result is calculated m 3 \u003d 3 780.

We have lost the calculation m4 = LCM (m3, a4) = LCM (3780, 250). Diemo behind this very algorithm. We take m 4 \u003d 94500.

NOK chotirioh numbers z think butt dorivnyuє 94500.

Suggestion: NOK (140, 9, 54, 250) = 94500.

Like bachite, the billing looks awkward, but to end up laborious. To spare the hour, you can go another way.

Appointment 4

We suggest you the next algorithm:

let's put these numbers into simple multipliers;
to the creation of multipliers of the first number, we add multipliers, which are daily, to the creation of another number;
up to the one removed at the previous stage, the creation is added daily multipliers of the third number, etc.;
otrimaniy tvir will be the smallest common multiple of all numbers from mind.

butt 8

It is necessary to know the LCM of the five numbers 84, 6, 48, 7, 143.

Solution

We decompose all five numbers into prime factors: 84 = 2 2 3 7, 6 = 2 3, 48 = 2 2 2 2 3, 7, 143 = 11 13. Forgive the numbers, for example, the number 7 in simple multipliers are not laid out. Such numbers zbіgayutsya zі svoїmy razladannyam on simple multipliers.

Now we take additional prime factors 2 , 2 , 3 and 7 of the number 84 and add factors of another number to them. We spread the number 6 on 2 and 3. Qi multipliers already є at the creation of the first number. Otzhe, їх we omit.

We continue to add daily multipliers. Let's move on to 48, with the addition of prime factors such as 2 and 2. Let's add a simple multiplier 7 on the fourth number and multipliers 11 and 13 on the fifth. Minor: 2 2 2 2 3 7 11 13 = 48048. Tse i є naimensha zagalnіst multiplicity of five vihіdnyh numbers.

Suggestion: NOK (84, 6, 48, 7, 143) = 48,048.

The value of the smallest common multiple of negative numbers

In order to know the least significant multiple of negative numbers, it is necessary to replace the first number by numbers with opposite signs, and then we will carry out the calculation for guidance by algorithms.

butt 9

LCM(54, -34) = LCM(54, 34) and LCM(-622, -46, -54, -888) = LCM(622, 46, 54, 888).

Such dії admissible y zv'azku s tim, sho yakscho accept, sho aі − a- Extended numbers,
then the number of multiples a zbіgaєtsya from the number of multiples − a.

butt 10

It is necessary to correct the LCM of negative numbers − 145 і − 45 .

Solution

Zrobimo change numbers − 145 і − 45 on protracted numbers 145 і 45 . Now, after the algorithm, we calculate the LCM (145, 45) = 145 45: GCD (145, 45) = 145 45: 5 = 1305, prioritizing the GCD behind the Euclidean algorithm.

It is taken into account that the NOC of numbers is 145 and − 45 one 1 305 .

Suggestion: LCM (− 145 , − 45) = 1 305 .

How did you remember the pardon in the text, be kind, see it and press Ctrl + Enter

Lesson 16

Qile: send the understanding of the smallest common multiple; form the name of the smallest common multiple; to practice the newcomer's task in an algebraic way; repeat the arithmetic mean.

Information for the reader

Turn back the respect of the students to a different szmist viraziv: “the highest multiple of the number”, “the least of the highest multiple of the number”.

The value of the smallest common multiple of the number of numbers:

1. Verify that more of these numbers are divided by other numbers.

2. As it continues, the same number will be the smallest overall multiple of all numbers.

3. If it doesn’t last, then revise it so that you don’t agree on the decision of the number of doubled, tripled, etc.

4. So pereverat until that hour, the docks are known to be the smallest number, so that the skin of the smallest numbers can be divided.

II way

2. Write a breakdown of one of the numbers (better write down the largest number at once).

If the numbers are mutually simple, then the smallest extreme multiple of these numbers will be їхнє tvіr.

Hid lesson

I. Organizational moment

II. Usny rahunok

1. Gra “I am the most important”.

15, 67, 38, 560, 435, 226, 1000, 539, 3255.

Clap at the valley, as the number is a multiple of 2.

Write down if the number is a multiple of 5.

Stomp your feet as the number is a multiple of 10.

Why were you splashing, squeaking and blunting your feet at the same time?

2. Name all simple numbers that satisfy the inconsistencies 20< х < 50.

3. What is bigger, how many sums of numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9? (Sum. Dobutok cost 0, and sum cost 45.)

4. Name a random number, written after the additional digits 1, 7, 5, 8, a multiple of 2, 5, 3. (1578, 1875, 1515.)

5. Marini had a whole apple, two halves and two quarters. How many apples did she have? (3.)

III. Individual work

(Date the task of learning, as if they pardoned themselves from independent robots, allowing them to speed up the records in class zoshit.)

1 card

a) 20 and 30; b) 8 and 9; c) 24 and 36.

2. Write down two numbers, for which the biggest sleeper will be the number: a) 5; b) 8.

a) 22 and 33; b) 24 and 30; c) 45 and 9; d) 15 and 35.

2 cards

1. Find out all the bedrooms of numbers and the seat of their largest sleep:

a) 30 and 40; b) 6 and 15; c) 28 and 42.

Name a pair of mutually prime numbers, for example.

2. Write down two numbers, for which the biggest sleeper will be the number: a) 3; b) 9.

3. Find the largest possible length of given numbers:

a) 33 and 44; b) 18 and 24; c) 36 and 9; d) 20 and 25.

IV. Informed by those lesson

Today at the lesson, it is obvious that such a lesser multiple of the numbers is such a way to know.

V. Introduction of new material

(The task is written on the board.)

Read the order.

From one pier to another walk two boats. Repair the work at once about the 8th anniversary of the wound. The first boat on the flight back and forth is 2 years old, and the other one is 3 years old.

After the smallest hour, the boats will again lean on the first pier, and how many flights will break the leather boat in the same hour?

How many times for the extraction of boats will be stranded on the first pier, and when will you come?

The joking hour can be extended without too much і by 2 і by 3, so it can be multiples of 2 and 3.

Let's write numbers that are multiples of 2 and 3:

Numbers multiple of 2: 2, 4, 6 , 8, 10, 12 , 14, 16, 18 , 20, 22, 24 .

Numbers that are multiples of 3: 3, 6 , 9, 12 , 15, 18 , 21, 24 .

Add a capital multiple of 2 and 3.

Name the smallest multiple of 2 and 3. (The smallest multiple is the number 6.)

Also, after 6 years, after a cob of robots, two boats lean on the first pier at the same time.

How many flights per hour to build a skin boat? (1 - 3 flights, 2 - 2 flights.)

How many times dobu and boats hit the first pier? (4 times.)

When are you going to see? (About the 14th year, 20th year, about the 2nd year of the night, about the 8th wound.)

Appointment. The smallest natural number, which is similar to the skin seen natural number, is called the smallest common multiple.

Signature: LCM (2; 3) = 6.

The smallest multiple of a number can be known and not written down as a multiple of a number.

For which requirement:

1. Lay out these numbers into prime factors.

2. Write the layout of one of the numbers (shorter for the largest).

3. Supplement the given layout with these multipliers from the layout of other numbers, as they did not reach the written layout.

4. Calculate the deductions of tvir.

Find the smallest possible multiple of numbers:

a) 75 and 60; b) 180, 45 and 60; c) 12 and 35.

The back of the head needs to be re-verified, so that it does not last longer on other numbers.

If so, then the greater will be the smallest major multiple of these numbers.

Then let's make it clear that these numbers are mutually forgiven.

If so, then the least extreme multiple will be tvіr tsikh numbers.

a) 75 is not divisible by 60 and the numbers 75 and 60 are not mutually simple, either

It’s better to immediately write down not the layout of the number 75, but the number itself.

b) The number 180 is divided by i by 45, i by 60, also,

NOC (180; 45; 60) = 180.

c) Qi numbers are mutually simple, also LCM (12; 35) = 420.

VI. Fizkulthvilinka

VII. Work on jobs

1. - Put in a short note.

(Three boxes had 160 kg of apples in the warehouse. The first box had 15 kg less, the second box had 15 kg less, the second box had 2 times more, the third box had less. How many kg of apples did the skin box have?)

Solve the problem using the algebra method.

(At the doshka, she is in zoshita.)

What is acceptable for x? Why? (Skіlki kg of apples in III box. Take less for x more often.)

So, what can you say about the second box? (2x (kg) apples in box II.)

Skilki will be in the first box? (2x - 15 (kg) apples in I box.)

What can equalize? (3 boxes have a total of 160 kg of apples.)

1) Let x (kg) - apples in III box,

2x (kg) - apples in the second box,

2x - 15 (kg) - apples in I box.

Knowing that in 3 boxes of a total of 160 kg of apples we store equal:

x + 2x + 2x - 15 = 160

x = 35; 35 kg of apples in box III.

2) 35 2 \u003d 70 (kg) - apples in the second box.

3) 70 - 15 = 55 (kg) - apples in the first box.

What do you need to do, first write down the order? (To write down the statement, it is necessary to read the food.)

Name the nutritional task. (How many kg of apples did the skin box have?)

Oskіlki we wrote a report explaining to dіy, vіdpovіd we will write down briefly.

(Vіdpoіd: 55 kg, 70 kg, 35 kg)

2. No. 184 side. 30 (bіlya doshki that in zoshita).

Read the order.

What needs to be done, what needs to be given for food? (Know the LCM of the numbers 45 and 60.)

45 = 3 3 5

60 = 2 5 2 3

NOC (45; 60) \u003d 60 3 \u003d 180, also 180 m.

(Vіdpovіd: 180 m)

VIII. Fixing the woven material

1. No. 179 side. 30 (bіlya doshki that in zoshita).

Find out the simple multipliers of the smallest common multiple and the largest common multiple of numbers a and b.

a) LCM (a; c) = 3 5 7

GCD (a; c) = 5.

b) LCM (a; c) = 2 2 3 3 5 7

GCD (a; c) = 2 2 3.

2. No. 180 (a, b) side. 30 (with reported comments).

a) LCM (a; b) = 2 3 3 3 5 2 5 = 2700.

b) Oskіlki b dilitsya a, then NOK, will be the number b itself.

LCM (a; b) = 2 3 3 5 7 7 = 4410.

IX. Repetition of woven material

1. - How to know the arithmetic mean of a number of numbers? (Know the sum of these numbers; subtracting the result, subtract by the number of numbers.)

No. 198 side. 32 (on the dosh and in the zoshites).

(3,8 + 4,2 + 3,5 + 4,1) : 4 = 3,9

2. No. 195 side. 32 (independently).

How else can you write privately two numbers? (Looking at the fraction.)