Mathematical Expressions and tasks require a lot of additional knowledge. NOC is one of the main ones, especially often used in The topic is studied in high school, and it is not particularly difficult to understand material; a person familiar with powers and the multiplication table will not have difficulty identifying the necessary numbers and discovering 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 the short name adopted for the designation, collected from the first letters.

Ways to get a number

The method of multiplying numbers is not always suitable for finding the LCM; it is much better suited for simple single-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 use prime, single- or double-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 a number 21, there is simply no smaller number.

Example No. 2

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

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

The second stage involves working with already obtained data. Each of the numbers received must participate in calculating the final result. For each factor, the largest number of occurrences is taken from the original numbers. NOC is total number, therefore, the factors from the numbers must be repeated in it, every single one, even those that are present in one copy. Both initial numbers contain the numbers 2, 3 and 5, in different degrees, 7 is present in only one case.

To calculate the final result, you need to take each number in the largest of the powers represented into the equation. All that remains is to multiply and get the answer; if filled out correctly, the task fits into two steps without explanation:

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

2) NOC = 6300.

That’s the whole problem, if you try to calculate the required number by multiplication, then the answer will definitely not be correct, since 300 * 1260 = 378,000.

Examination:

6300 / 300 = 21 - correct;

6300 / 1260 = 5 - correct.

The correctness of the result obtained is determined by checking - dividing the LCM by both initial 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 reduce fractions to a common denominator. What is usually studied in grades 5-6 of secondary school. It is also additionally a common divisor for all multiples, if such conditions are present 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 there are in the task, but the complexity does not increase.

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

1) 250 = 25 * 10 = 5 2 *5 * 2 = 5 3 * 2 - this example describes 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 necessary to mention all the factors, in this case 2, 5, 3 are given - for all these numbers it is necessary to determine the maximum degree.

Attention: all factors must be brought to the point of complete simplification, if possible, decomposed to the level of single digits.

Examination:

1) 3000 / 250 = 12 - correct;

2) 3000 / 600 = 5 - true;

3) 3000 / 1500 = 2 - correct.

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

Another way

In mathematics, many things are connected, many things 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 digit numbers. A table is compiled into which the multiplicand is entered vertically, the multiplier horizontally, and the product is indicated in the intersecting cells of the column. You can reflect the table using a line, take a number and write down the results of multiplying this number by integers, from 1 to infinity, sometimes 3-5 points are enough, the second and subsequent numbers undergo the same computational process. Everything happens until a common multiple is found.

Given the numbers 30, 35, 42, you need to find the LCM connecting 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 NOC. Among the processes associated with this calculation there is also the largest common divisor, which is calculated according to similar principles and is often found in neighboring problems. The difference is small, but quite significant, LCM involves calculating the number that is divided by all given initial values, and GCD involves calculating the largest value by which the original numbers are divided.

But many natural numbers are also 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 by a whole (for 12 these are 1, 2, 3, 4, 6 and 12) are called divisors of numbers. Divisor of a natural number a- is a natural number that divides given number a without a trace. A natural number that has more than two divisors is called composite .

Please note that the numbers 12 and 36 have common factors. These numbers are: 1, 2, 3, 4, 6, 12. The greatest divisor of these numbers is 12. The common divisor of these two numbers a And b- this is the number by which both given numbers are divided without remainder a And b.

Common multiples several numbers is a 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 common multiples there is always a smallest one, in this case it is 90. This number is called the smallestcommon multiple (CMM).

The LCM is always a natural number that 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 of the LCM( m, n).

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

So, Chebyshev function. And:

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

What follows from the distribution law 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 connection with the LCM:

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

Where p 1 ,...,p k- various prime numbers, and d 1 ,...,d k And e 1 ,...,e k— non-negative integers (they can be zeros if the corresponding prime is not in the expansion).

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

In other words, the LCM decomposition contains all prime factors included in at least one of the decompositions of numbers a, b, and the largest of the two exponents of this multiplier is taken.

Example:

Calculating the least common multiple of several numbers can be reduced to several sequential 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 (the product of the factors of the desired product) into the factors of the desired product large number from the given ones), and then add factors from the expansion of other numbers that do not appear in the first number or appear in it fewer times;

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

Any two or more natural numbers have their own NOC. 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) are supplemented with the factor 3 (the number 21), the resulting product (84) will be the smallest number, which is divisible by 21 and 28.

The prime factors of the largest number 30 are supplemented by the factor 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 is a multiple of all given numbers.

The numbers 2,3,11,37 are prime numbers, 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 = 2 2 2 3 3 7,

2) write down the powers of all prime factors:

504 = 2 2 2 3 3 7 = 2 3 3 2 7 1,

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

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

5) multiply these powers.

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

Solution. 168 = 2 2 2 3 7 = 2 3 3 1 7 1,

180 = 2 2 3 3 5 = 2 2 3 2 5 1,

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

We write down the greatest powers of all prime divisors and multiply them:

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

The material presented below is a logical continuation of the theory from the article entitled LCM - least common multiple, definition, examples, connection between LCM and GCD. Here we will talk about finding the least common multiple (LCM), And Special attention Let's focus on solving examples. First, we will show how the LCM of two numbers is calculated using the GCD of these numbers. Next, we'll look at finding the least common multiple by factoring numbers into prime factors. After this, we will focus on finding the LCM of three or more numbers, and also pay attention to calculating the LCM of negative numbers.

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Calculating Least Common Multiple (LCM) via GCD

One way to find the least common multiple is based on the relationship between LCM and GCD. Existing connection between LCM and GCD allows you to calculate the least common multiple of two positive integers using a known greatest common divisor. The corresponding formula is LCM(a, b)=a b:GCD(a, b) . Let's look at examples of finding the LCM using the given formula.

Example.

Find the least common multiple of two numbers 126 and 70.

Solution.

In this example a=126 , b=70 . Let us use the connection between LCM and GCD, expressed by the formula LCM(a, b)=a b:GCD(a, b). That is, first we have to find the greatest common divisor of the numbers 70 and 126, after which we can calculate the LCM of these numbers using the written formula.

Let's find GCD(126, 70) using the Euclidean algorithm: 126=70·1+56, 70=56·1+14, 56=14·4, therefore, GCD(126, 70)=14.

Now we find the required least common multiple: GCD(126, 70)=126·70:GCD(126, 70)= 126·70:14=630.

Answer:

LCM(126, 70)=630 .

Example.

What is LCM(68, 34) equal to?

Solution.

Because 68 is divisible by 34, then GCD(68, 34)=34. Now we calculate the least common multiple: GCD(68, 34)=68·34:GCD(68, 34)= 68·34:34=68.

Answer:

LCM(68, 34)=68 .

Note that the previous example fits the following rule for finding the LCM for positive integers a and b: if the number a is divisible by b, then the least common multiple of these numbers is a.

Finding the LCM by factoring numbers into prime factors

Another way to find the least common multiple is based on factoring numbers into prime factors. If you compose a product from all the prime factors of given numbers, and then exclude from this product all the common prime factors present in the decompositions of the given numbers, then the resulting product will be equal to the least common multiple of the given numbers.

The stated rule for finding the LCM follows from the equality LCM(a, b)=a b:GCD(a, b). Indeed, the product of numbers a and b is equal to the product of all factors involved in the expansion of numbers a and b. In turn, gcd(a, b) equal to the product all prime factors that are simultaneously present in the expansions of numbers a and b (as described in the section on finding GCD using the expansion of numbers into prime factors).

Let's give an example. Let us know that 75=3·5·5 and 210=2·3·5·7. Let's compose the product from all the factors of these expansions: 2·3·3·5·5·5·7 . Now from this product we exclude all the factors present in both the expansion of the number 75 and the expansion of the number 210 (these factors are 3 and 5), then the product will take the form 2·3·5·5·7. The value of this product is equal to the least common multiple of 75 and 210, that is, NOC(75, 210)= 2·3·5·5·7=1,050.

Example.

Factor the numbers 441 and 700 into prime factors and find the least common multiple of these numbers.

Solution.

Let's factor the numbers 441 and 700 into prime factors:

We get 441=3·3·7·7 and 700=2·2·5·5·7.

Now let’s create a product from all the factors involved in the expansion of these numbers: 2·2·3·3·5·5·7·7·7. Let us exclude from this product all factors that are simultaneously present in both expansions (there is only one such factor - this is the number 7): 2·2·3·3·5·5·7·7. Thus, LCM(441, 700)=2·2·3·3·5·5·7·7=44 100.

Answer:

NOC(441, 700)= 44 100 .

The rule for finding the LCM using factorization of numbers into prime factors can be formulated a little differently. If the missing factors from the expansion of number b are added to the factors from the expansion of the number a, then the value of the resulting product will be equal to the least common multiple of the numbers a and b.

For example, let's take the same numbers 75 and 210, their decompositions into prime factors are as follows: 75=3·5·5 and 210=2·3·5·7. To the factors 3, 5 and 5 from the expansion of the number 75 we add the missing factors 2 and 7 from the expansion of the number 210, we obtain the product 2·3·5·5·7, the value of which is equal to LCM(75, 210).

Example.

Find the least common multiple of 84 and 648.

Solution.

We first obtain the decompositions of the numbers 84 and 648 into prime factors. They look like 84=2·2·3·7 and 648=2·2·2·3·3·3·3. To the factors 2, 2, 3 and 7 from the expansion of the number 84 we add the missing factors 2, 3, 3 and 3 from the expansion of the number 648, we obtain the product 2 2 2 3 3 3 3 7, which is equal to 4 536 . Thus, the desired least common multiple of 84 and 648 is 4,536.

Answer:

LCM(84, 648)=4,536 .

Finding the LCM of three or more numbers

The least common multiple of three or more numbers can be found by sequentially finding the LCM of two numbers. Let us recall the corresponding theorem, which gives a way to find the LCM of three or more numbers.

Theorem.

Let positive integer numbers a 1 , a 2 , …, a k be given, the least common multiple m k of these numbers is found by sequentially calculating m 2 = LCM(a 1 , a 2) , m 3 = LCM(m 2 , a 3) , … , m k = LCM(m k−1 , a k) .

Let's consider the application of this theorem using the example of finding the least common multiple of four numbers.

Example.

Find the LCM of four numbers 140, 9, 54 and 250.

Solution.

In this example, a 1 =140, a 2 =9, a 3 =54, a 4 =250.

First we find m 2 = LOC(a 1 , a 2) = LOC(140, 9). To do this, using the Euclidean algorithm, we determine GCD(140, 9), we have 140=9·15+5, 9=5·1+4, 5=4·1+1, 4=1·4, therefore, GCD(140, 9)=1 , from where GCD(140, 9)=140 9:GCD(140, 9)= 140·9:1=1,260. That is, m 2 =1 260.

Now we find m 3 = LOC (m 2 , a 3) = LOC (1 260, 54). Let's calculate it through GCD(1 260, 54), which we also determine using the Euclidean algorithm: 1 260=54·23+18, 54=18·3. Then gcd(1,260, 54)=18, from which gcd(1,260, 54)= 1,260·54:gcd(1,260, 54)= 1,260·54:18=3,780. That is, m 3 =3 780.

All that remains is to find m 4 = LOC(m 3, a 4) = LOC(3 780, 250). To do this, we find GCD(3,780, 250) using the Euclidean algorithm: 3,780=250·15+30, 250=30·8+10, 30=10·3. Therefore, GCM(3,780, 250)=10, whence GCM(3,780, 250)= 3 780 250: GCD(3 780, 250)= 3,780·250:10=94,500. That is, m 4 =94,500.

So the least common multiple of the original four numbers is 94,500.

Answer:

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

In many cases, it is convenient to find the least common multiple of three or more numbers using prime factorizations of the given numbers. In this case, you should adhere to the following rule. The least common multiple of several numbers is equal to the product, which is composed as follows: the missing factors from the expansion of the second number are added to all the factors from the expansion of the first number, the missing factors from the expansion of the third number are added to the resulting factors, and so on.

Let's look at an example of finding the least common multiple using prime factorization.

Example.

Find the least common multiple of the five numbers 84, 6, 48, 7, 143.

Solution.

First, we obtain decompositions of these numbers into prime factors: 84=2·2·3·7, 6=2·3, 48=2·2·2·2·3, 7 (7 is a prime number, it coincides with its decomposition into prime factors) and 143=11·13.

To find the LCM of these numbers, to the factors of the first number 84 (they are 2, 2, 3 and 7), you need to add the missing factors from the expansion of the second number 6. The decomposition of the number 6 does not contain missing factors, since both 2 and 3 are already present in the decomposition of the first number 84. Next, to the factors 2, 2, 3 and 7 we add the missing factors 2 and 2 from the expansion of the third number 48, we get a set of factors 2, 2, 2, 2, 3 and 7. There will be no need to add multipliers to this set in the next step, since 7 is already contained in it. Finally, to the factors 2, 2, 2, 2, 3 and 7 we add the missing factors 11 and 13 from the expansion of the number 143. We get the product 2·2·2·2·3·7·11·13, which is equal to 48,048.

But many natural numbers are also 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 by a whole (for 12 these are 1, 2, 3, 4, 6 and 12) are called divisors of numbers. Divisor of a natural number a- is a natural number that divides a given number a without a trace. A natural number that has more than two divisors is called composite .

Please note that the numbers 12 and 36 have common factors. These numbers are: 1, 2, 3, 4, 6, 12. The greatest divisor of these numbers is 12. The common divisor of these two numbers a And b- this is the number by which both given numbers are divided without remainder a And b.

Common multiples several numbers is a 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 common multiples there is always a smallest one, in this case it is 90. This number is called the smallestcommon multiple (CMM).

The LCM is always a natural number that 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 of the LCM( m, n).

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

So, Chebyshev function. And:

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 connection with the LCM:

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

Where p 1 ,...,p k- various prime numbers, and d 1 ,...,d k And e 1 ,...,e k— non-negative integers (they can be zeros if the corresponding prime is not in the expansion).

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

In other words, the LCM decomposition contains all prime factors included in at least one of the decompositions of numbers a, b, and the largest of the two exponents of this multiplier is taken.

Example:

Calculating the least common multiple of several numbers can be reduced to several sequential 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 decomposition (the product of the factors of the largest number of the given ones) to the factors of the desired product, and then add factors from the decomposition of other numbers that do not appear in the first number or appear in it fewer times;

— the resulting product of prime factors will be the LCM of the 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) are 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 are supplemented by the factor 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 is a multiple of all given numbers.

The numbers 2,3,11,37 are prime numbers, 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 = 2 2 2 3 3 7,

2) write down the powers of all prime factors:

504 = 2 2 2 3 3 7 = 2 3 3 2 7 1,

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

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

5) multiply these powers.

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

Solution. 168 = 2 2 2 3 7 = 2 3 3 1 7 1,

180 = 2 2 3 3 5 = 2 2 3 2 5 1,

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

We write down the greatest powers of all prime divisors and multiply them:

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

How to find LCM (least common multiple)

A common multiple of two integers is an integer that is evenly divisible by both given numbers without leaving a remainder.

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

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 numbers 6 and 9 will be equal to 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 factoring the original numbers into prime factors.
After decomposition, it is necessary to cross out identical numbers from the resulting series of prime factors. The remaining numbers of the first number will be a multiplier for the second, and the remaining numbers of the second will be a multiplier for the first.

Example for numbers 75 and 60.
The least common multiple of the numbers 75 and 60 can be found without writing down the multiples of these numbers in a row. To do this, let’s factor 75 and 60 into simple factors:
75 = 3 * 5 * 5, a
60 = 2 * 2 * 3 * 5 .
As you can see, factors 3 and 5 appear in both rows. We mentally “cross out” them.
Let us write down the remaining factors included in the expansion of each of these numbers. When decomposing the number 75, we are left with the number 5, and when decomposing the number 60, we are left with 2 * 2
This means that in order 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 multiply the numbers remaining from the expansion of 60 (this is 2 * 2) by 75. That is, for ease of understanding , we say that we are multiplying “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 the LCM for the numbers 12, 16, 24
In this case, our actions will be somewhat more complicated. But first, as always, let’s factorize all the numbers
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 sequentially go through its factors, crossing them out if in at least one of the other rows of numbers we encounter the same factor that has not yet been crossed out.

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

Step 2. In the prime factors of the 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 actions are 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. This means that the finding of the LOC is completed. All that remains is to calculate its value.
For the number 12, take the remaining factors of the number 16 (next 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 method allows you to do it faster. However, both methods of finding the LCM are correct.