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Tasks for GCD and LCM of numbers The work of a 6th grade student of the MKOU "Kamyshovskaya OOSh" Lantsinova Aisa Supervisor Goryaeva Zoya Erdnigoryaevna, teacher of mathematics p. Kamyshovo, 2013

An example of finding the GCD of the numbers 50, 75 and 325. 1) Let's decompose the numbers 50, 75 and 325 into prime factors. 50= 2 ∙ 5 ∙ 5 75= 3 ∙ 5 ∙ 5 325= 5 ∙ 5 ∙ 13 50= 2 ∙ 5 ∙ 5 75= 3 ∙ 5 ∙ 5 325= 5 ∙ 5 ∙13 divide without a remainder the numbers a and b are called the greatest common divisor of these numbers.

An example of finding the LCM of the numbers 72, 99 and 117. 1) Let us factorize the numbers 72, 99 and 117. Write out the factors included in the expansion of one of the numbers 2 ∙ 2 ∙ 2 ∙ 3 ​​∙ 3 and add to them the missing factors of the remaining numbers. 2 ∙ 2 ∙ 2 ∙ 3 ​​∙ 3 ∙ 11 ∙ 13 3) Find the product of the resulting factors. 2 ∙ 2 ∙ 2 ∙ 3 ​​∙ 3 ∙ 11 ∙ 13= 10296 Answer: LCM (72, 99 and 117) = 10296 The least common multiple of natural numbers a and b is the smallest natural number that is a multiple of a and b.

A sheet of cardboard has the shape of a rectangle, the length of which is 48 cm and the width is 40 cm. This sheet must be cut without waste into equal squares. What are the largest squares that can be obtained from this sheet and how many? Solution: 1) S = a ∙ b is the area of ​​the rectangle. S \u003d 48 ∙ 40 \u003d 1960 cm². is the area of ​​the cardboard. 2) a - the side of the square 48: a - the number of squares that can be laid along the length of the cardboard. 40: a - the number of squares that can be laid across the width of the cardboard. 3) GCD (40 and 48) \u003d 8 (cm) - the side of the square. 4) S \u003d a² - the area of ​​\u200b\u200bone square. S \u003d 8² \u003d 64 (cm².) - the area of ​​\u200b\u200bone square. 5) 1960: 64 = 30 (number of squares). Answer: 30 squares with a side of 8 cm each. Tasks for GCD

The fireplace in the room must be laid out with finishing tiles in the shape of a square. How many tiles will be needed for a 195 ͯ 156 cm fireplace and what are largest dimensions tiles? Solution: 1) S = 196 ͯ 156 = 30420 (cm ²) - S of the fireplace surface. 2) GCD (195 and 156) = 39 (cm) - side of the tile. 3) S = a² = 39² = 1521 (cm²) - area of ​​1 tile. 4) 30420: = 20 (pieces). Answer: 20 tiles measuring 39 ͯ 39 (cm). Tasks for GCD

A garden plot measuring 54 ͯ 48 m around the perimeter must be fenced off, for this, concrete pillars must be placed at regular intervals. How many poles must be brought for the site, and at what maximum distance from each other will the poles stand? Solution: 1) P = 2(a + b) – site perimeter. P \u003d 2 (54 + 48) \u003d 204 m. 2) GCD (54 and 48) \u003d 6 (m) - the distance between the pillars. 3) 204: 6 = 34 (pillars). Answer: 34 pillars, at a distance of 6 m. Tasks for GCD

Out of 210 burgundy, 126 white, 294 red roses, bouquets were collected, and in each bouquet the number of roses of the same color is equal. What is the largest number of bouquets made from these roses and how many roses of each color are in one bouquet? Solution: 1) GCD (210, 126 and 294) = 42 (bouquets). 2) 210: 42 = 5 (burgundy roses). 3) 126: 42 = 3 (white roses). 4) 294: 42 = 7 (red roses). Answer: 42 bouquets: 5 burgundy, 3 white, 7 red roses in each bouquet. Tasks for GCD

Tanya and Masha bought the same number of mailboxes. Tanya paid 90 rubles, and Masha paid 5 rubles. more. How much does one set cost? How many sets did each buy? Solution: 1) Masha paid 90 + 5 = 95 (rubles). 2) GCD (90 and 95) = 5 (rubles) - the price of 1 set. 3) 980: 5 = 18 (sets) - bought by Tanya. 4) 95: 5 = 19 (sets) - Masha bought. Answer: 5 rubles, 18 sets, 19 sets. Tasks for GCD

Three tourist boat trips start in the port city, the first of which lasts 15 days, the second - 20 and the third - 12 days. Returning to the port, the ships on the same day again go on a voyage. Motor ships left the port on all three routes today. In how many days will they sail together for the first time? How many trips will each ship make? Solution: 1) NOC (15.20 and 12) = 60 (days) - meeting time. 2) 60: 15 = 4 (voyages) - 1 ship. 3) 60: 20 = 3 (voyages) - 2 motor ship. 4) 60: 12 = 5 (voyages) - 3 motor ship. Answer: 60 days, 4 flights, 3 flights, 5 flights. Tasks for the NOC

Masha bought eggs for the Bear in the store. On the way to the forest, she realized that the number of eggs is divisible by 2,3,5,10 and 15. How many eggs did Masha buy? Solution: LCM (2;3;5;10;15) = 30 (eggs) Answer: Masha bought 30 eggs. Tasks for the NOC

It is required to make a box with a square bottom for stacking boxes measuring 16 ͯ 20 cm. What should be the shortest side of the square bottom to fit the boxes tightly into the box? Solution: 1) NOC (16 and 20) = 80 (boxes). 2) S = a ∙ b is the area of ​​1 box. S \u003d 16 ∙ 20 \u003d 320 (cm ²) - the area of ​​​​the bottom of 1 box. 3) 320 ∙ 80 = 25600 (cm ²) - square bottom area. 4) S \u003d a² \u003d a ∙ a 25600 \u003d 160 ∙ 160 - the dimensions of the box. Answer: 160 cm is the side of the square bottom. Tasks for the NOC

Along the road from point K there are power poles every 45 m. It was decided to replace these poles with others, placing them at a distance of 60 m from each other. How many poles were there and how many will they stand? Solution: 1) NOK (45 and 60) = 180. 2) 180: 45 = 4 - there were pillars. 3) 180: 60 = 3 - there were pillars. Answer: 4 pillars, 3 pillars. Tasks for the NOC

How many soldiers are marching on the parade ground if they march in formation of 12 people in a line and change into a column of 18 people in a line? Solution: 1) NOC (12 and 18) = 36 (people) - marching. Answer: 36 people. Tasks for the NOC

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

For example:

The number 12 is divisible by 1, by 2, by 3, by 4, by 6, by 12;

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

The numbers by which the number is divisible (for 12 it is 1, 2, 3, 4, 6 and 12) are called number divisors. Divisor of a natural number a is the natural number that divides the given number a without a trace. A natural number that has more than two factors is called composite .

Note that the numbers 12 and 36 have common divisors. These are the numbers: 1, 2, 3, 4, 6, 12. The largest divisor of these numbers is 12. The common divisor of these two numbers a and b is the number by which both given numbers are divisible without a remainder a and b.

common multiple several numbers is called the number that is divisible by each of these numbers. For example, the numbers 9, 18 and 45 have a common multiple of 180. But 90 and 360 are also their common multiples. Among all jcommon multiples, there is always the smallest one, in this case it is 90. This number is called leastcommon multiple (LCM).

LCM is always a natural number, which must be greater than the largest of the numbers for which it is defined.

Least common multiple (LCM). Properties.

Commutativity:

Associativity:

In particular, if and are coprime numbers , then:

Least common multiple of two integers m and n is a divisor of all other common multiples m and n. Moreover, the set of common multiples m,n coincides with the set of multiples for LCM( m,n).

The asymptotics for can be expressed in terms of some number-theoretic functions.

So, Chebyshev function. As well as:

This follows from the definition and properties of the Landau function g(n).

What follows from the law of distribution of prime numbers.

Finding the least common multiple (LCM).

NOC( a, b) can be calculated in several ways:

1. If the greatest common divisor is known, you can use its relationship with the LCM:

2. Let the canonical decomposition of both numbers into prime factors be known:

where p 1 ,...,p k- various prime numbers, a d 1 ,...,d k and e 1 ,...,ek are non-negative integers (they can be zero if the corresponding prime is not in the expansion).

Then LCM ( a,b) is calculated by the formula:

In other words, the LCM expansion contains all prime factors that are included in at least one of the number expansions a, b, and the largest of the two exponents of this factor is taken.

Example:

The calculation of the least common multiple of several numbers can be reduced to several successive calculations of the LCM of two numbers:

Rule. To find the LCM of a series of numbers, you need:

- decompose numbers into prime factors;

- transfer the largest expansion to the factors of the desired product (the product of the factors of the largest number of the given ones), and then add factors from the expansion of other numbers that do not occur in the first number or are in it a smaller number of times;

- the resulting product of prime factors will be LCM given numbers.

Any two or more natural numbers have their own LCM. If the numbers are not multiples of each other or do not have the same factors in the expansion, then their LCM is equal to the product of these numbers.

The prime factors of the number 28 (2, 2, 7) were supplemented with a factor of 3 (the number 21), the resulting product (84) will be the smallest number that is divisible by 21 and 28.

The prime factors of the largest number 30 were supplemented with a factor of 5 of the number 25, the resulting product 150 is greater than the largest number 30 and is divisible by all given numbers without a remainder. This is the smallest possible product (150, 250, 300...) that all given numbers are multiples of.

The numbers 2,3,11,37 are prime, so their LCM is equal to the product of the given numbers.

rule. To calculate the LCM of prime numbers, you need to multiply all these numbers together.

Another option:

To find the least common multiple (LCM) of several numbers you need:

1) represent each number as a product of its prime factors, for example:

504 \u003d 2 2 2 3 3 7,

2) write down the powers of all prime factors:

504 \u003d 2 2 2 3 3 7 \u003d 2 3 3 2 7 1,

3) write down all prime divisors (multipliers) of each of these numbers;

4) choose the largest degree of each of them, found in all expansions of these numbers;

5) multiply these powers.

Example. Find the LCM of numbers: 168, 180 and 3024.

Solution. 168 \u003d 2 2 2 3 7 \u003d 2 3 3 1 7 1,

180 \u003d 2 2 3 3 5 \u003d 2 2 3 2 5 1,

3024 = 2 2 2 2 3 3 3 7 = 2 4 3 3 7 1 .

We write out the largest powers of all prime divisors and multiply them:

LCM = 2 4 3 3 5 1 7 1 = 15120.

Students are given a lot of math assignments. Among them, very often there are tasks with the following formulation: there are two values. How to find the least common multiple of given numbers? It is necessary to be able to perform such tasks, since the acquired skills are used to work with fractions when different denominators. In the article, we will analyze how to find the LCM and the basic concepts.

Before finding the answer to the question of how to find the LCM, you need to define the term multiple. Most often, the wording of this concept is as follows: a multiple of some value A is a natural number that will be divisible by A without a remainder. So, for 4, 8, 12, 16, 20 and so on, up to the required limit.

In this case, the number of divisors for a particular value can be limited, and there are infinitely many multiples. There is also the same value for natural values. This is an indicator that is divided by them without a remainder. Having dealt with the concept of the smallest value for certain indicators, let's move on to how to find it.

Finding the NOC

The smallest multiple of two or more exponents is the smallest natural number, which is fully divisible by all given numbers.

There are several ways to find such a value. Let's consider the following methods:

1. If the numbers are small, then write in the line all divisible by it. Keep doing this until you find something in common among them. In the record, they are denoted by the letter K. For example, for 4 and 3, the smallest multiple is 12.
2. If these are large or you need to find a multiple for 3 or more values, then you should use a different technique here, which involves decomposing numbers into prime factors. First, lay out the largest of the indicated, then all the rest. Each of them has its own number of multipliers. As an example, let's decompose 20 (2*2*5) and 50 (5*5*2). For the smaller of them, underline the factors and add to the largest. The result will be 100, which will be the least common multiple of the above numbers.
3. When finding 3 numbers (16, 24 and 36) the principles are the same as for the other two. Let's expand each of them: 16 = 2*2*2*2, 24=2*2*2*3, 36=2*2*3*3. Only two deuces from the expansion of the number 16 were not included in the decomposition of the largest. We add them and get 144, which is the smallest result for the previously indicated numerical values.

Now we know what is the general technique for finding the smallest value for two, three or more values. However, there are also private methods, helping to search for NOCs, if the previous ones do not help.

How to find GCD and NOC.

Private Ways of Finding

As with any mathematical section, there are special cases of finding LCMs that help in specific situations:

• if one of the numbers is divisible by the others without a remainder, then the lowest multiple of these numbers is equal to it (NOC 60 and 15 is equal to 15);
• Coprime numbers do not have common prime divisors. Their smallest value is equal to the product of these numbers. Thus, for the numbers 7 and 8, this will be 56;
• the same rule works for other cases, including special ones, which can be read about in specialized literature. This should also include cases of decomposition of composite numbers, which are the subject of separate articles and even Ph.D. dissertations.

Special cases are less common than standard examples. But thanks to them, you can learn how to work with fractions of varying degrees of complexity. This is especially true for fractions., where there are different denominators.

Some examples

Let's look at a few examples, thanks to which you can understand the principle of finding the smallest multiple:

1. We find LCM (35; 40). We lay out first 35 = 5*7, then 40 = 5*8. We add 8 to the smallest number and get the NOC 280.
2. NOC (45; 54). We lay out each of them: 45 = 3*3*5 and 54 = 3*3*6. We add the number 6 to 45. We get the NOC equal to 270.
3. Well, the last example. There are 5 and 4. There are no simple multiples for them, so the least common multiple in this case will be their product, equal to 20.

Thanks to examples, you can understand how the NOC is located, what are the nuances and what is the meaning of such manipulations.

Finding the NOC is much easier than it might seem at first. For this, both a simple expansion and the multiplication of simple values ​​\u200b\u200bto each other are used.. The ability to work with this section of mathematics helps with further study mathematical topics, especially fractions of varying degrees of complexity.

Do not forget to periodically solve examples with different methods, this develops the logical apparatus and allows you to remember numerous terms. Learn methods for finding such an indicator and you will be able to work well with the rest of the mathematical sections. Happy learning math!

Video

This video will help you understand and remember how to find the least common multiple.

Mathematical expressions and tasks require a lot of additional knowledge. NOC is one of the main ones, especially often used in the topic. The topic is studied in high school, while it is not particularly difficult to understand material, it will not be difficult for a person familiar with powers and the multiplication table to select the necessary numbers and find the result.

Definition

A common multiple is a number that can be completely divided into two numbers at the same time (a and b). Most often, this number is obtained by multiplying the original numbers a and b. The number must be divisible by both numbers at once, without deviations.

NOC is a short name, which is taken from the first letters.

Ways to get a number

To find the LCM, the method of multiplying numbers is not always suitable, it is much better suited for simple one-digit or two-digit numbers. It is customary to divide into factors, the larger the number, the more factors there will be.

Example #1

For the simplest example, schools usually take simple, one-digit or two-digit numbers. For example, you need to solve the following task, find the least common multiple of the numbers 7 and 3, the solution is quite simple, just multiply them. As a result, there is the number 21, there is simply no smaller number.

Example #2

The second option is much more difficult. The numbers 300 and 1260 are given, finding the LCM is mandatory. To solve the task, the following actions are assumed:

Decomposition of the first and second numbers into the simplest factors. 300 = 2 2 * 3 * 5 2 ; 1260 = 2 2 * 3 2 * 5 * 7. The first stage has been completed.

The second stage involves working with the already obtained data. Each of the received numbers must participate in the calculation of the final result. For each multiplier, the most big number occurrences. LCM is a common number, so the factors from the numbers must be repeated in it to the last, even those that are present in one instance. Both initial numbers have in their composition the numbers 2, 3 and 5, in different degrees, 7 is only in one case.

To calculate the final result, you need to take each number in the largest of their represented powers, into the equation. It remains only to multiply and get the answer, with the correct filling, the task fits into two steps without explanation:

1) 300 = 2 2 * 3 * 5 2 ; 1260 = 2 2 * 3 2 *5 *7.

2) NOK = 6300.

That's the whole task, if you try to calculate the desired number by multiplying, then the answer will definitely not be correct, since 300 * 1260 = 378,000.

Examination:

6300 / 300 = 21 - true;

6300 / 1260 = 5 is correct.

The correctness of the result is determined by checking - dividing the LCM by both original numbers, if the number is an integer in both cases, then the answer is correct.

What does NOC mean in mathematics

As you know, there is not a single useless function in mathematics, this one is no exception. The most common purpose of this number is to bring fractions to a common denominator. What is usually studied in grades 5-6 high school. It is also additionally a common divisor for all multiples, if such conditions are in the problem. Such an expression can find a multiple not only of two numbers, but also of a much larger number - three, five, and so on. How more numbers- the more actions in the task, but the complexity of this does not increase.

For example, given the numbers 250, 600 and 1500, you need to find their total LCM:

1) 250 = 25 * 10 = 5 2 * 5 * 2 = 5 3 * 2 - this example describes the factorization in detail, without reduction.

2) 600 = 60 * 10 = 3 * 2 3 *5 2 ;

3) 1500 = 15 * 100 = 33 * 5 3 *2 2 ;

In order to compose an expression, it is required to mention all factors, in this case 2, 5, 3 are given - for all these numbers it is required to determine the maximum degree.

Attention: all multipliers must be brought to full simplification, if possible, decomposing to the level of single digits.

Examination:

1) 3000 / 250 = 12 - true;

2) 3000 / 600 = 5 - true;

3) 3000 / 1500 = 2 is correct.

This method does not require any tricks or genius level abilities, everything is simple and clear.

Another way

In mathematics, a lot is connected, a lot can be solved in two or more ways, the same goes for finding the least common multiple, LCM. The following method can be used in the case of simple two-digit and single digits. A table is compiled in which the multiplier is entered vertically, the multiplier horizontally, and the product is indicated in the intersecting cells of the column. You can reflect the table by means of a line, a number is taken and the results of multiplying this number by integers are written in a row, from 1 to infinity, sometimes 3-5 points are enough, the second and subsequent numbers are subjected to the same computational process. Everything happens until a common multiple is found.

Given the numbers 30, 35, 42, you need to find the LCM that connects all the numbers:

1) Multiples of 30: 60, 90, 120, 150, 180, 210, 250, etc.

2) Multiples of 35: 70, 105, 140, 175, 210, 245, etc.

3) Multiples of 42: 84, 126, 168, 210, 252, etc.

It is noticeable that all the numbers are quite different, the only common number among them is 210, so it will be the LCM. Among the processes associated with this calculation, there is also the largest common divisor, which is calculated according to similar principles and is often encountered in neighboring problems. The difference is small, but significant enough, the LCM involves the calculation of a number that is divisible by all given initial values, and the GCM involves the calculation greatest value by which the original numbers are divisible.

How to find LCM (least common multiple)

The common multiple of two integers is the integer that is evenly divisible by both given numbers without remainder.

The least common multiple of two integers is the smallest of all integers that is divisible evenly and without remainder by both given numbers.

Method 1. You can find the LCM, in turn, for each of the given numbers, writing out in ascending order all the numbers that are obtained by multiplying them by 1, 2, 3, 4, and so on.

Example for numbers 6 and 9.
We multiply the number 6, sequentially, by 1, 2, 3, 4, 5.
We get: 6, 12, 18 , 24, 30
We multiply the number 9, sequentially, by 1, 2, 3, 4, 5.
We get: 9, 18 , 27, 36, 45
As you can see, the LCM for the numbers 6 and 9 will be 18.

This method is convenient when both numbers are small and it is easy to multiply them by a sequence of integers. However, there are cases when you need to find the LCM for two-digit or three-digit numbers, and also when there are three or even more initial numbers.

Method 2. You can find the LCM by decomposing the original numbers into prime factors.
After decomposition, it is necessary to cross out the same numbers from the resulting series of prime factors. The remaining numbers of the first number will be the factor for the second, and the remaining numbers of the second number will be the factor for the first.

Example for the number 75 and 60.
The least common multiple of the numbers 75 and 60 can be found without writing out multiples of these numbers in a row. To do this, we decompose 75 and 60 into prime factors:
75 = 3 * 5 * 5, and
60 = 2 * 2 * 3 * 5 .
As you can see, the factors 3 and 5 occur in both rows. Mentally we "cross out" them.
Let's write down the remaining factors included in the expansion of each of these numbers. When decomposing the number 75, we left the number 5, and when decomposing the number 60, we left 2 * 2
So, to determine the LCM for the numbers 75 and 60, we need to multiply the remaining numbers from the expansion of 75 (this is 5) by 60, and the numbers remaining from the expansion of the number 60 (this is 2 * 2) multiply by 75. That is, for ease of understanding , we say that we multiply "crosswise".
75 * 2 * 2 = 300
60 * 5 = 300
This is how we found the LCM for the numbers 60 and 75. This is the number 300.

Example. Determine LCM for numbers 12, 16, 24
In this case, our actions will be somewhat more complicated. But, first, as always, we decompose all numbers into prime factors
12 = 2 * 2 * 3
16 = 2 * 2 * 2 * 2
24 = 2 * 2 * 2 * 3
To correctly determine the LCM, we select the smallest of all numbers (this is the number 12) and successively go through its factors, crossing them out if at least one of the other rows of numbers has the same multiplier that has not yet been crossed out.

Step 1 . We see that 2 * 2 occurs in all series of numbers. We cross them out.
12 = 2 * 2 * 3
16 = 2 * 2 * 2 * 2
24 = 2 * 2 * 2 * 3

Step 2. In prime factors number 12, only the number 3 remains. But it is present in the prime factors of the number 24. We cross out the number 3 from both rows, while no action is expected for the number 16.
12 = 2 * 2 * 3
16 = 2 * 2 * 2 * 2
24 = 2 * 2 * 2 * 3

As you can see, when decomposing the number 12, we "crossed out" all the numbers. So the finding of the NOC is completed. It remains only to calculate its value.
For the number 12, we take the remaining factors from the number 16 (the closest in ascending order)
12 * 2 * 2 = 48
This is the NOC

As you can see, in this case, finding the LCM was somewhat more difficult, but when you need to find it for three or more numbers, this way allows you to do it faster. However, both ways of finding the LCM are correct.