How to find the least common multiple of 6 cells. How to find the least common multiple of numbers

To use the preview of presentations, create a Google account (account) and sign in: https://accounts.google.com

Slides captions:

Math lesson in 6th grade. Mathematics teacher GBOU secondary school №539 Dmitry Vadimovich Labzin. Least common multiple.

oral work. 1. Calculate: a) ? ? 2. It is known that Come up with correct statements using the terms: “is a divisor”, “is divisible”, “is a multiple”. Which of them are synonyms? 3. Is it possible to assert that the numbers a, b and c are multiples of 14 if: - Find the quotient of dividing the number a by 14, the number b by 14.

In writing. 2. Find some common multiples of 15 and 30. Solution. Multiples of 15:15; thirty; 45; 60; 75; 90… Multiples of 30:30; 60; 90…Common multiples: 30; 60; 90. - What is the least common multiple of the numbers 15 and 30. - The number 30. - Try to formulate what number is called the least common multiple of two natural numbers a and b? The least common multiple of natural numbers a and b is the smallest natural number that is a multiple of both a and b. - Tell me, please, is the considered method of finding the NOC convenient? - Why? LCM(15;30) = 30. They write:

2. Numbers are given: - Think about how you can find the least common multiple of the numbers a and b? Algorithm. 1. Decompose these numbers into prime factors; 2. Write out the decomposition of one of them; 3. Add the missing factors from the expansion of another number; 4. Find the resulting work.

Example 1. Find LCM (32;25). Decision. Let's decompose the numbers 32 and 25 into prime factors. ; - What can be said about the numbers 32 and 25? The least common multiple of coprime numbers is equal to their product. Example 2. Find the LCM of numbers 12; fifteen; 20; 60. Decision. If among the numbers there is one that is divisible by all the others, then this is the LCM of these numbers. - What did you notice?

Numbers given: 15 and 30. Multiples of 15: 15; thirty; 45; 60; 75; 90… Multiples of 30:30; 60; 90… Least common multiple: 30. That's interesting! Multiples of 30: 30; 60; 90… Each multiple of LCM (a; b) is a common multiple of a and b and, conversely, each of their common multiple is a multiple of LCM (a; b).

To understand how to calculate the LCM, you should first determine the meaning of the term "multiple".

A multiple of A is a natural number that is divisible by A without remainder. Thus, 15, 20, 25, and so on can be considered multiples of 5.

There can be a limited number of divisors of a particular number, but there are an infinite number of multiples.

A common multiple of natural numbers is a number that is divisible by them without a remainder.

How to find the least common multiple of numbers

The least common multiple (LCM) of numbers (two, three or more) is the smallest natural number that is evenly divisible by all these numbers.

To find the NOC, you can use several methods.

For small numbers, it is convenient to write out in a line all the multiples of these numbers until a common one is found among them. Multiples are denoted in the record with a capital letter K.

For example, multiples of 4 can be written like this:

K(4) = (8,12, 16, 20, 24, ...)

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

So, you can see that the least common multiple of the numbers 4 and 6 is the number 24. This entry is performed as follows:

LCM(4, 6) = 24

If the numbers are large, find the common multiple of three or more numbers, then it is better to use another way to calculate the LCM.

To complete the task, it is necessary to decompose the proposed numbers into prime factors.

First you need to write out the expansion of the largest of the numbers in a line, and below it - the rest.

In the expansion of each number, there may be a different number of factors.

For example, let's factor the numbers 50 and 20 into prime factors.

In the expansion of the smaller number, one should underline the factors that are missing in the expansion of the first largest number, and then add them to it. In the presented example, a deuce is missing.

Now we can calculate the least common multiple of 20 and 50.

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

Thus, the product of the prime factors of the larger number and the factors of the second number, which are not included in the decomposition of the larger number, will be the least common multiple.

To find the LCM of three or more numbers, all of them should be decomposed into prime factors, as in the previous case.

As an example, you can find the least common multiple of the numbers 16, 24, 36.

36 = 2 * 2 * 3 * 3

24 = 2 * 2 * 2 * 3

16 = 2 * 2 * 2 * 2

Thus, only two deuces from the decomposition of sixteen were not included in the factorization of a larger number (one is in the decomposition of twenty-four).

Thus, they need to be added to the decomposition of a larger number.

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

There are special cases of determining the least common multiple. So, if one of the numbers can be divided without a remainder by another, then the larger of these numbers will be the least common multiple.

For example, NOCs of twelve and twenty-four would be twenty-four.

If it is necessary to find the least common multiple of coprime numbers that do not have the same divisors, then their LCM will be equal to their product.

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

Let's continue the discussion about the least common multiple that we started in the LCM - Least Common Multiple, Definition, Examples section. In this topic, we will look at ways to find the LCM for three numbers or more, we will analyze the question of how to find the LCM of a negative number.

Calculation of the least common multiple (LCM) through gcd

We have already established the relationship between the least common multiple and the greatest common divisor. Now let's learn how to define the LCM through the GCD. First, let's figure out how to do this for positive numbers.

Definition 1

You can find the least common multiple through the greatest common divisor using the formula LCM (a, b) \u003d a b: GCD (a, b) .

Example 1

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

Decision

Let's take a = 126 , b = 70 . Substitute the values ​​in the formula for calculating the least common multiple through the greatest common divisor LCM (a, b) = a · b: GCD (a, b) .

Finds the GCD of the numbers 70 and 126. For this we need the Euclid algorithm: 126 = 70 1 + 56 , 70 = 56 1 + 14 , 56 = 14 4 , hence gcd (126 , 70) = 14 .

Let's calculate the LCM: LCM (126, 70) = 126 70: GCD (126, 70) = 126 70: 14 = 630.

Answer: LCM (126, 70) = 630.

Example 2

Find the nok of the numbers 68 and 34.

Decision

GCD in this case is easy to find, since 68 is divisible by 34. Calculate the least common multiple using the formula: LCM (68, 34) = 68 34: GCD (68, 34) = 68 34: 34 = 68.

Answer: LCM(68, 34) = 68.

In this example, we used the rule for finding the least common multiple of positive integers a and b: if the first number is divisible by the second, then the LCM of these numbers will be equal to the first number.

Finding the LCM by Factoring Numbers into Prime Factors

Now let's look at a way to find the LCM, which is based on the decomposition of numbers into prime factors.

Definition 2

To find the least common multiple, we need to perform a number of simple steps:

• we make up the product of all prime factors of numbers for which we need to find the LCM;
• we exclude all prime factors from their obtained products;
• the product obtained after eliminating the common prime factors will be equal to the LCM of the given numbers.

This way of finding the least common multiple is based on the equality LCM (a , b) = a b: GCD (a , b) . If you look at the formula, it will become clear: the product of the numbers a and b is equal to the product of all factors that are involved in the expansion of these two numbers. In this case, the GCD of two numbers is equal to the product of all prime factors that are simultaneously present in the factorizations of these two numbers.

Example 3

We have two numbers 75 and 210 . We can factor them out like this: 75 = 3 5 5 and 210 = 2 3 5 7. If you make the product of all the factors of the two original numbers, you get: 2 3 3 5 5 5 7.

If we exclude the factors common to both numbers 3 and 5, we get a product of the following form: 2 3 5 5 7 = 1050. This product will be our LCM for the numbers 75 and 210.

Example 4

Find the LCM of numbers 441 and 700 , decomposing both numbers into prime factors.

Decision

Let's find all the prime factors of the numbers given in the condition:

441 147 49 7 1 3 3 7 7

700 350 175 35 7 1 2 2 5 5 7

We get two chains of numbers: 441 = 3 3 7 7 and 700 = 2 2 5 5 7 .

The product of all the factors that participated in the expansion of these numbers will look like: 2 2 3 3 5 5 7 7 7. Let's find the common factors. This number is 7 . We exclude it from the general product: 2 2 3 3 5 5 7 7. It turns out that NOC (441 , 700) = 2 2 3 3 5 5 7 7 = 44 100.

Answer: LCM (441 , 700) = 44 100 .

Let us give one more formulation of the method for finding the LCM by decomposing numbers into prime factors.

Definition 3

Previously, we excluded from the total number of factors common to both numbers. Now we will do it differently:

• Let's decompose both numbers into prime factors:
• add to the product of the prime factors of the first number the missing factors of the second number;
• we get the product, which will be the desired LCM of two numbers.

Example 5

Let's go back to the numbers 75 and 210 , for which we already looked for the LCM in one of the previous examples. Let's break them down into simple factors: 75 = 3 5 5 and 210 = 2 3 5 7. To the product of factors 3 , 5 and 5 number 75 add the missing factors 2 and 7 numbers 210 . We get: 2 3 5 5 7 . This is the LCM of the numbers 75 and 210.

Example 6

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

Decision

Let's decompose the numbers from the condition into prime factors: 84 = 2 2 3 7 and 648 = 2 2 2 3 3 3 3. Add to the product of the factors 2 , 2 , 3 and 7 numbers 84 missing factors 2 , 3 , 3 and
3 numbers 648 . We get the product 2 2 2 3 3 3 3 7 = 4536 . This is the least common multiple of 84 and 648.

Answer: LCM (84, 648) = 4536.

Finding the LCM of three or more numbers

Regardless of how many numbers we are dealing with, the algorithm of our actions will always be the same: we will consistently find the LCM of two numbers. There is a theorem for this case.

Theorem 1

Suppose we have integers a 1 , a 2 , … , a k. NOC m k of these numbers is found in 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 look at how the theorem can be applied to specific problems.

Example 7

You need to calculate the least common multiple of the four numbers 140 , 9 , 54 and 250 .

Decision

Let's introduce the notation: a 1 \u003d 140, a 2 \u003d 9, a 3 \u003d 54, a 4 \u003d 250.

Let's start by calculating m 2 = LCM (a 1 , a 2) = LCM (140 , 9) . Let's use the Euclidean algorithm to calculate the GCD of the numbers 140 and 9: 140 = 9 15 + 5 , 9 = 5 1 + 4 , 5 = 4 1 + 1 , 4 = 1 4 . We get: GCD(140, 9) = 1, LCM(140, 9) = 140 9: GCD(140, 9) = 140 9: 1 = 1260. Therefore, m 2 = 1 260 .

Now let's calculate according to the same algorithm m 3 = LCM (m 2 , a 3) = LCM (1 260 , 54) . In the course of calculations, we get m 3 = 3 780.

It remains for us to calculate m 4 \u003d LCM (m 3, a 4) \u003d LCM (3 780, 250) . We act according to the same algorithm. We get m 4 \u003d 94 500.

The LCM of the four numbers from the example condition is 94500 .

Answer: LCM (140, 9, 54, 250) = 94,500.

As you can see, the calculations are simple, but quite laborious. To save time, you can go the other way.

Definition 4

We offer you the following algorithm of actions:

• decompose all numbers into prime factors;
• to the product of the factors of the first number, add the missing factors from the product of the second number;
• add the missing factors of the third number to the product obtained at the previous stage, etc.;
• the resulting product will be the least common multiple of all numbers from the condition.

Example 8

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

Decision

Let's 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 . Prime numbers, which is the number 7, cannot be factored into prime factors. Such numbers coincide with their decomposition into prime factors.

Now let's take the product of the prime factors 2, 2, 3 and 7 of the number 84 and add to them the missing factors of the second number. We have decomposed the number 6 into 2 and 3. These factors are already in the product of the first number. Therefore, we omit them.

We continue to add the missing multipliers. We turn to the number 48, from the product of prime factors of which we take 2 and 2. Then we add a simple factor of 7 from the fourth number and factors of 11 and 13 of the fifth. We get: 2 2 2 2 3 7 11 13 = 48,048. This is the least common multiple of the five original numbers.

Answer: LCM (84, 6, 48, 7, 143) = 48,048.

Finding the Least Common Multiple of Negative Numbers

In order to find the least common multiple of negative numbers, these numbers must first be replaced by numbers with the opposite sign, and then the calculations should be carried out according to the above algorithms.

Example 9

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

Such actions are permissible due to the fact that if it is accepted that a and − a- opposite numbers
then the set of multiples a coincides with the set of multiples of a number − a.

Example 10

It is necessary to calculate the LCM of negative numbers − 145 and − 45 .

Decision

Let's change the numbers − 145 and − 45 to their opposite numbers 145 and 45 . Now, using the algorithm, we calculate the LCM (145 , 45) = 145 45: GCD (145 , 45) = 145 45: 5 = 1 305 , having previously determined the GCD using the Euclid algorithm.

We get that the LCM of numbers − 145 and − 45 equals 1 305 .

Answer: LCM (− 145 , − 45) = 1 305 .

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

Lesson 16

Goals: introduce the concepts of the least common multiple; to form the skill of finding the least common multiple; develop the skill of solving problems in an algebraic way; repeat the arithmetic mean.

Information for the teacher

Draw students' attention to the different meanings of the expressions: "common multiple of numbers", "least common multiple of numbers".

Finding the least common multiple of several numbers:

1. Check if the larger of the given numbers is divisible by the rest of the numbers.

2. If divisible, then this number will be the least common multiple of all given numbers.

3. If it is not divisible, then check whether a doubled larger number, tripled, etc. will not be divisible by other numbers.

4. So check until you find the smallest number that is divisible by each of the other numbers.

II way

2. Write the expansion of one of the numbers (it is better to immediately write down the largest number).

If the numbers are coprime, then the least common multiple of these numbers will be their product.

During the classes

I. Organizational moment

II. Verbal counting

1. The game "I am the most attentive."

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

Clap your hands if the number is a multiple of 2.

Write down if the number is a multiple of 5.

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

Why were you clapping, squeaking and stamping your feet at the same time?

2. Name all prime numbers that satisfy inequality 20< х < 50.

3. Which is greater, the product or the sum of these numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9? (Sum. The product is 0 and the sum is 45.)

4. What is a four-digit number written using the numbers 1, 7, 5, 8, a multiple of 2, 5, 3. (1578, 1875, 1515.)

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

III. Individual work

(Give a task to students who made mistakes in independent work, allowing them to use the notes in the class notebook.)

1 card

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

2. Write down two numbers for which the greatest common divisor is the number: a) 5; b) 8.

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

2 card

1. Find all common divisors of numbers and underline their greatest common divisor:

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

Name a pair of relatively prime numbers, if any.

2. Write down two numbers for which the greatest common divisor is the number: a) 3; b) 9.

3. Find the greatest common divisor of these numbers:

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

IV. Lesson topic message

Today in the lesson we will find out what the least common multiple of numbers is and how to find it.

V. Learning new material

(The problem is written on the board.)

Read the task.

Two boats run from one pier to the other. They start work at the same time at 8 o'clock in the morning. The first boat spends 2 hours on a round trip, and the second - 3 hours.

What is the shortest time after which both boats will again be at the first pier, and how many trips will each boat make during this time?

How many times a day will these boats meet at the first pier, and at what time will this happen?

The desired time must be divisible without a remainder by both 2 and 3, that is, it must be a multiple of 2 and 3.

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

Numbers that are multiples 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 .

Underline the common multiples of 2 and 3.

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

This means that 6 hours after the start of work, two boats will simultaneously be at the first pier.

How many trips will each boat make during this time? (1 - 3 flights, 2 - 2 flights.)

How many times per day will these boats meet at the first pier? (4 times.)

What time will this take place? (At 2 pm, 8 pm, 2 am, 8 am.)

Definition. The smallest natural number that is divisible by each of the published natural numbers is called the least common multiple.

Notation: LCM (2; 3) = 6.

The least common multiple of numbers can be found without writing multiples in a row.

For this you need:

1. Decompose all numbers into prime factors.

2. Write the expansion of one of the numbers (better than the largest).

3. Supplement this expansion with those factors from the expansion of other numbers that were not included in the written expansion.

4. Calculate the resulting product.

Find the least common multiple of the numbers:

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

First you need to check if the larger number is divisible by other numbers.

If yes, then the larger number will be the least common multiple of those numbers.

Then determine if the given numbers are coprime.

If yes, then the least common multiple will be the product of these numbers.

a) 75 is not divisible by 60, and the numbers 75 and 60 are not coprime, then

It is better to immediately write down not the decomposition of the number 75, but this number itself.

b) The number 180 is divisible by both 45 and 60, therefore,

NOC (180; 45; 60) = 180.

c) These numbers are relatively prime, so LCM (12; 35) = 420.

VI. Physical education minute

VII. Working on a task

1. - Make a task on a short note.

(There were 160 kg of apples in the warehouse in three boxes. In the first box, 15 kg less, in the second, in the second, 2 times more than in the third. How many kg of apples were in each box?)

Solve the problem using the algebraic method.

(At the blackboard and in notebooks.)

What do we take for x? Why? (How many kg of apples are in box III. It is better to take a smaller number for x.)

Then what can be said about box II? (2x (kg) apples in box II.)

How many will be in box 1? (2x - 15 (kg) apples in I box.)

What can be used to create an equation? (There are only 160 kg of apples in 3 boxes.)

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

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

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

Knowing that there are only 160 kg of apples in 3 boxes, we make the equation:

x + 2x + 2x - 15 = 160

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

2) 35 2 = 70 (kg) - apples in box II.

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

What should be done before writing the answer to the problem? (To write down the answer, you need to read the question of the problem.)

Name the question of the task. (How many kg of apples were in each box?)

Since we wrote a detailed explanation of the actions, we will write down the answer briefly.

(Answer: 55 kg, 70 kg, 35 kg.)

2. No. 184 p. 30 (at the blackboard and in notebooks).

Read the task.

What needs to be done to answer the question of the problem? (Find 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, which means 180 m.

(Answer: 180 m.)

VIII. Consolidation of the studied material

1. No. 179 p. 30 (at the blackboard and in notebooks).

Find the prime factorization of the least common multiple and the greatest common divisor of the 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) p. 30 (with detailed commentary).

a) LCM (a; b) \u003d 2 3 3 3 5 2 5 \u003d 2700.

b) Since b is divisible by a, then LCM will be the number b itself.

LCM (a; b) \u003d 2 3 3 5 7 7 \u003d 4410.

IX. Repetition of the studied material

1. - How to find the arithmetic mean of several numbers? (Find the sum of these numbers; divide the result by the number of numbers.)

No. 198 p. 32 (on the board and in notebooks).

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

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

How else can you write the quotient of two numbers? (As a fraction.)

X. Independent work

Record intermediate answers.

Option I. No. 125 (1-2 lines) p. 22, No. 222 (a-c) p. 36, No. 186 (a, b) p. 31.

Option II. No. 125 (3-4 lines) p. 22, No. 186 (c, d) p. 31, No. 222 (e) p. 36.

XI. Summing up the lesson

What is the common multiple of these numbers?

What is the least common multiple of these numbers?

How to find the least common multiple of given numbers?

Homework

No. 202 (a, b, find GCD and NOC), No. 204 p. 32, No. 206 (a) p. 33, No. 145 (a) p. 24.

Individual task: No. 201 p. 32.

Topic: "Least common multiple", Grade 6, UMK Vilenkin N.Ya.

Lesson type: "discovery" of new knowledge.

Basic goals.

Construct the definition of the least common multiple, the algorithm for finding the LCM. To form the ability to find the NOC.

train ability

To the use of the concepts of prime and composite number;

Signs of divisibility by 2, 3, 5, 9, 10:

Different ways to find the NOC:

Algorithms for finding the intersection and union of sets;

3) Train the ability to factorize.

I Self-determination to activity.

Let's do a workout. Children are divided into groups according to options. The first take a card with a task and announce to their group:

1st - sign of divisibility by 2;

2nd - a sign of divisibility by 3;

3rd - a sign of divisibility by 5;

4th - a sign of divisibility by 9;

5th - a sign of divisibility by 10;

6th - a sign of divisibility by 2 ..

The numbers appear on the presentation screen: 51, 22, 37, 191, 163, 88, 47, 133, 152, 202, 403, 75, 507, 609, 708, and the children must write down in their notebook those numbers that are determined by the assignment (or rise from their place, if the sign given to them can be applied to the number)

Guys, why do you need to know the signs of divisibility? (for factoring numbers)

II. Knowledge update

What classes can all natural numbers be divided into according to the number of divisors? (into prime and compound and 1)

What numbers are called prime? (numbers with only two divisors)

List some prime numbers) (2,3,5,7,9,11,13,17,…)

Tell me, for what problems is the decomposition into prime factors used? (finding the greatest common divisor (learned in previous lessons))

What is the algorithm for finding GCD? (an algorithm for finding GCD is formulated using factorization)

Find the greatest common divisor of 18 and 24?

How did you find it. Children are called with different ways of finding the GCD (by writing all the divisors of numbers, through decomposition into prime factors).

Compare the GCD to each of the numbers.

III. Statement of the educational task and fixation of the difficulty of 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)

Common multiples of these numbers: 72. 144

Name the number 72 (Least common multiple of these numbers: 72)

So, formulate the topic of today's lesson (least common multiple)

What is the purpose of the lesson? (learn to find the NOC)

We found the LCM by the selection method, but what other method can be used to find the LCM? (by the method of decomposition into prime factors)

What is the essence of this method?

IV. Building a project to get out of difficulty

Together with the children, an algorithm for finding the NOC is compiled.

For this you need:

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

V. Primary consolidation in external speech.

Workbook, p. 28 No. 3 abc

Tasks are performed with commenting in accordance with the derived algorithm according to the above proposed scheme.

VI. Independent work with self-test according to the standard

Students perform independently No. 181 (abcg)

Decided right

Errors are corrected, their causes are identified and spoken out.

At this time, students who correctly completed the task can additionally do No. 183

VII. Inclusion in the knowledge system and repetition.

Students who made mistakes in independent work at this stage perform No. 4 RT (workbook, p. 29) to find the least common multiple.

The rest of the students decide in groups No. 193, 161, 192

The captains present solutions.

VIII. Reflection of activity. (outcome of the lesson).

- What is the common multiple of these numbers?

What is the least common multiple of these numbers?

How to find the least common multiple?

Students on a segment from 0 to 1 put up a figure depicting the level of understanding of a new topic, for example

IX. Homework.

P.7 pp. 29-30, No. 202, 204, 206(ab) additionally (optional) No. 209 with a presentation at the next lesson.