Common multiple of numbers. How to find the least common multiple of two numbers

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 factor, the largest number of occurrences is taken from the original numbers. 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 of 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. The 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-digit numbers. 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 greatest common divisor, which is calculated according to similar principles and is often encountered in neighboring problems. The difference is small, but significant enough, LCM involves the calculation of a number that is divisible by all given initial values, and GCD assumes the calculation of the largest value by which the initial numbers are divided.

Let's start studying the least common multiple of two or more numbers. In the section, we will give a definition of the term, consider a theorem that establishes a relationship between the least common multiple and the greatest common divisor, and give examples of solving problems.

Common multiples - definition, examples

In this topic, we will be interested only in common multiples of integers other than zero.

Definition 1

Common multiple of integers is an integer that is a multiple of all given numbers. In fact, it is any integer that can be divided by any of the given numbers.

The definition of common multiples refers to two, three, or more integers.

Example 1

According to the definition given above for the number 12, the common multiples are 3 and 2. Also the number 12 will be a common multiple of the numbers 2 , 3 and 4 . The numbers 12 and -12 are common multiples of the numbers ±1, ±2, ±3, ±4, ±6, ±12.

At the same time, the common multiple for the numbers 2 and 3 will be the numbers 12 , 6 , − 24 , 72 , 468 , − 100 010 004 and a number of any others.

If we take numbers that are divisible by the first number of a pair and not divisible by the second, then such numbers will not be common multiples. So, for the numbers 2 and 3, the numbers 16 , − 27 , 5009 , 27001 will not be common multiples.

0 is a common multiple of any set of non-zero integers.

If we recall the property of divisibility with respect to opposite numbers, then it turns out that some integer k will be a common multiple of these numbers in the same way as the number - k. This means that common divisors can be either positive or negative.

Is it possible to find an LCM for all numbers?

The common multiple can be found for any integers.

Example 2

Suppose we are given k integers a 1 , a 2 , … , a k. The number that we get during the multiplication of numbers a 1 a 2 … a k according to the divisibility property, it will be divided by each of the factors that were included in the original product. This means that the product of the numbers a 1 , a 2 , … , a k is the least common multiple of these numbers.

How many common multiples can these integers have?

A group of integers can have a large number of common multiples. In fact, their number is infinite.

Example 3

Suppose we have some number k . Then the product of the numbers k · z , where z is an integer, will be a common multiple of the numbers k and z . Given that the number of numbers is infinite, then the number of common multiples is infinite.

Least Common Multiple (LCM) - Definition, Symbol and Examples

Recall the concept of the smallest number from a given set of numbers, which we considered in the Comparison of Integers section. With this concept in mind, we formulate the definition of the least common multiple, which has the greatest practical significance among all common multiples.

Definition 2

Least common multiple of given integers is the least positive common multiple of these numbers.

The least common multiple exists for any number of given numbers. The abbreviation NOK is the most commonly used to designate a concept in the reference literature. Shorthand for Least Common Multiple for Numbers a 1 , a 2 , … , a k will look like LCM (a 1 , a 2 , … , a k).

Example 4

The least common multiple of 6 and 7 is 42. Those. LCM(6, 7) = 42. The least common multiple of four numbers - 2 , 12 , 15 and 3 will be equal to 60 . Shorthand will be LCM (- 2 , 12 , 15 , 3) ​​= 60 .

Not for all groups of given numbers, the least common multiple is obvious. Often it has to be calculated.

Relationship between NOC and NOD

The least common multiple and the greatest common divisor are related. The relationship between concepts is established by the theorem.

Theorem 1

The least common multiple of two positive integers a and b is equal to the product of the numbers a and b divided by the greatest common divisor of the numbers a and b , that is, LCM (a , b) = a b: GCD (a , b) .

Proof 1

Suppose we have some number M which is a multiple of numbers a and b . If the number M is divisible by a , there is also some integer z , under which the equality M = a k. According to the definition of divisibility, if M is also divisible by b, so then a k divided by b.

If we introduce a new notation for gcd (a , b) as d, then we can use the equalities a = a 1 d and b = b 1 · d . In this case, both equalities will be coprime numbers.

We have already established above that a k divided by b. Now this condition can be written as follows:
a 1 d k divided by b 1 d, which is equivalent to the condition a 1 k divided by b 1 according to the properties of divisibility.

According to the property of relatively prime numbers, if a 1 and b 1 are mutually prime numbers, a 1 not divisible by b 1 despite the fact that a 1 k divided by b 1, then b 1 should share k.

In this case, it would be appropriate to assume that there is a number t, for which k = b 1 t, and since b1=b:d, then k = b: d t.

Now instead of k put into equality M = a k expression of the form b: d t. This allows us to come to equality M = a b: d t. At t=1 we can get the least positive common multiple of a and b , equal a b: d, provided that the numbers a and b positive.

So we have proved that LCM (a , b) = a b: GCD (a,b).

Establishing a connection between LCM and GCD allows you to find the least common multiple through the greatest common divisor of two or more given numbers.

Definition 3

The theorem has two important consequences:

• multiples of the least common multiple of two numbers are the same as common multiples of those two numbers;
• the least common multiple of coprime positive numbers a and b is equal to their product.

It is not difficult to substantiate these two facts. Any common multiple of M numbers a and b is defined by the equality M = LCM (a, b) t for some integer value t. Since a and b are coprime, then gcd (a, b) = 1, therefore, LCM (a, b) = a b: gcd (a, b) = a b: 1 = a b.

Least common multiple of three or more numbers

In order to find the least common multiple of several numbers, you must successively find the LCM of two numbers.

Theorem 2

Let's pretend that a 1 , a 2 , … , a k are some positive integers. To calculate the LCM m k these numbers, we need to sequentially calculate m 2 = LCM(a 1 , a 2) , m 3 = NOC(m 2 , a 3) , … , m k = NOC(m k - 1 , a k) .

Proof 2

The first corollary of the first theorem discussed in this topic will help us to prove the correctness of the second theorem. Reasoning is built according to the following algorithm:

• common multiples of numbers a 1 and a 2 coincide with multiples of their LCM, in fact, they coincide with multiples of the number m2;
• common multiples of numbers a 1, a 2 and a 3 m2 and a 3 m 3;
• common multiples of numbers a 1 , a 2 , … , a k coincide with common multiples of numbers m k - 1 and a k, therefore, coincide with multiples of the number m k;
• due to the fact that the smallest positive multiple of the number m k is the number itself m k, then the least common multiple of the numbers a 1 , a 2 , … , a k is m k.

So we have proved the theorem.

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With the concepts of the greatest common divisor (GCD) and the least common multiple (LCM), high school students meet in the sixth grade. This topic is always difficult to master. Children often confuse these concepts, do not understand why they need to be studied. Recently, in the popular science literature, there are separate statements that this material should be excluded from the school curriculum. I think that this is not entirely true, and it is necessary to study it, if not in the classroom, then during extracurricular time in the classroom of the school component, as it contributes to the development of the logical thinking of schoolchildren, increasing the speed of computational operations, and the ability to solve problems using beautiful methods.

When studying the topic "Addition and subtraction of fractions with different denominators" we teach children to find the common denominator of two or more numbers. For example, you need to add the fractions 1/3 and 1/5. Students can easily find a number that is divisible without a remainder by 3 and 5. This number is 15. Indeed, if the numbers are small, then their common denominator is easy to find, knowing the multiplication table well. One of the guys notices that this number is the product of the numbers 3 and 5. The children have the opinion that you can always find a common denominator for numbers in this way. For example, subtract the fractions 7/18 and 5/24. Let's find the product of the numbers 18 and 24. It is equal to 432. We have already received a large number, and if further calculations need to be made (especially for examples for all actions), then the probability of an error increases. But the found least common multiple of the numbers (LCM), which in this case is equivalent to the least common denominator (LCD) - the number 72 - will greatly facilitate calculations and lead to a faster solution of the example, and thereby save the time allotted for completing this task, which plays an important role in the performance of the final test, control work, especially during the final certification.

When studying the topic "Reduction of fractions", you can move successively by dividing the numerator and denominator of the fraction by the same natural number, using the signs of divisibility of numbers, eventually obtaining an irreducible fraction. For example, you need to reduce the fraction 128/344. We first divide the numerator and denominator of the fraction by the number 2, we get the fraction 64/172. Once again, we divide the numerator and denominator of the resulting fraction by 2, we get the fraction 32/86. Divide once again the numerator and denominator of the fraction by 2, we get the irreducible fraction 16/43. But fraction reduction can be done much easier if we find the greatest common divisor of the numbers 128 and 344. GCD (128, 344) = 8. Dividing the numerator and denominator of the fraction by this number, we immediately get an irreducible fraction.

Show children different ways to find the greatest common divisor (GCD) and least common multiple (LCM) of numbers. In simple cases, it is convenient to find the greatest common divisor (GCD) and least common multiple (LCM) of numbers by simple enumeration. As the numbers get larger, prime factors can be used. The sixth grade textbook (author N.Ya. Vilenkin) shows the following method for finding the greatest common divisor (GCD) of numbers. Let's decompose the numbers into prime factors:

• 16 = 2*2*2*2
• 120 = 2*2*2*3*5

Then, from the factors included in the expansion of one of these numbers, we cross out those that are not included in the expansion of the other number. The product of the remaining factors will be the greatest common divisor of these numbers. In this case, this number is 8. From my own experience, I was convinced that it is more understandable for children if we underline the same factors in the expansions of numbers, and then in one of the expansions we find the product of the underlined factors. This is the greatest common divisor of these numbers. In the sixth grade, children are active and inquisitive. You can set them the following task: try to find the greatest common divisor of the numbers 343 and 287 in the described way. It is not immediately clear how to factor them into prime factors. And here you can tell them about the wonderful method invented by the ancient Greeks, which allows you to search for the greatest common divisor (GCD) without decomposing into prime factors. This method of finding the greatest common divisor was first described in Euclid's Elements. It is called the Euclid algorithm. It consists in the following: First, divide the larger number by the smaller one. If there is a remainder, then divide the smaller number by the remainder. If the remainder is obtained again, then divide the first remainder by the second. So continue to divide until the remainder is zero. The last divisor is the greatest common divisor (GCD) of these numbers.

Let's return to our example and, for clarity, write the solution in the form of a table.

So gcd(344,287) = 7

And how to find the least common multiple (LCM) of the same numbers? Is there some way for this that does not require a preliminary decomposition of these numbers into prime factors? It turns out there is, and a very simple one at that. We need to multiply these numbers and divide the product by the greatest common divisor (GCD) we found. In this example, the product of the numbers is 98441. Divide it by 7 and get the number 14063. LCM(343,287) = 14063.

One of the difficult topics in mathematics is the solution of word problems. We need to show students how to use the concepts of "Greatest Common Divisor (GCD)" and "Least Common Multiple (LCM)" to solve problems that are sometimes difficult to solve in the usual way. Here it is appropriate to consider with students, along with the tasks proposed by the authors of the school textbook, old and entertaining tasks that develop children's curiosity and increase interest in studying this topic. Skillful possession of these concepts allows students to see a beautiful solution to a non-standard problem. And if the child’s mood rises after solving a good problem, this is a sign of successful work.

Thus, the study at school of such concepts as "Greatest Common Divisor (GCD)" and "Least Common Multiple (LCD)" of numbers

Allows you to save time allotted for the execution of work, which leads to a significant increase in the volume of completed tasks;

Increases the speed and accuracy of performing arithmetic operations, which leads to a significant reduction in the number of allowable computational errors;

Allows you to find beautiful ways to solve non-standard text problems;

Develops the curiosity of students, broadens their horizons;

Creates the prerequisites for the education of a versatile creative personality.

The largest natural number by which the numbers a and b are divisible without remainder is called greatest common divisor these numbers. Denote GCD(a, b).

Consider finding the GCD using the example of two natural numbers 18 and 60:

324 , 111 and 432

Let's decompose the numbers into prime factors:

324 = 2×2×3×3×3×3

111 = 3×37

432 = 2×2×2×2×3×3×3

Delete from the first number, the factors of which are not in the second and third numbers, we get:

2 x 2 x 2 x 2 x 3 x 3 x 3 = 3

As a result of GCD( 324 , 111 , 432 )=3

Finding GCD with Euclid's Algorithm

The second way to find the greatest common divisor using Euclid's algorithm. Euclid's algorithm is the most efficient way to find GCD, using it you need to constantly find the remainder of the division of numbers and apply recurrent formula.

Recurrent formula for GCD, gcd(a, b)=gcd(b, a mod b), where a mod b is the remainder of dividing a by b.

Euclid's algorithm

Example Find the Greatest Common Divisor of Numbers 7920 and 594

Let's find GCD( 7920 , 594 ) using the Euclid algorithm, we will calculate the remainder of the division using a calculator.

• 7920 mod 594 = 7920 - 13 × 594 = 198
• 594 mod 198 = 594 - 3 × 198 = 0

As a result, we get GCD( 7920 , 594 ) = 198

Least common multiple

In order to find a common denominator when adding and subtracting fractions with different denominators, you need to know and be able to calculate least common multiple(NOC).

A multiple of the number "a" is a number that is itself divisible by the number "a" without a remainder.

Numbers that are multiples of 8 (that is, these numbers will be divided by 8 without a remainder): these are the numbers 16, 24, 32 ...

Multiples of 9: 18, 27, 36, 45…

There are infinitely many multiples of a given number a, in contrast to the divisors of the same number. Divisors - a finite number.

A common multiple of two natural numbers is a number that is evenly divisible by both of these numbers..

Least common multiple(LCM) of two or more natural numbers is the smallest natural number that is itself divisible by each of these numbers.

How to find the NOC

LCM can be found and written in two ways.

The first way to find the LCM

This method is usually used for small numbers.

1. We write the multiples for each of the numbers in a line until there is a multiple that is the same for both numbers.
2. A multiple of the number "a" is denoted by a capital letter "K".

Example. Find LCM 6 and 8.

The second way to find the LCM

This method is convenient to use to find the LCM for three or more numbers.

The number of identical factors in the expansions of numbers can be different.

You can also formalize finding the least common multiple (LCM) as follows. Let's find the LCM (12, 16, 24) .

24 = 2 2 2 3

As you can see from the expansion of numbers, all factors of 12 are included in the expansion of 24 (the largest of the numbers), so we add only one 2 from the expansion of the number 16 to the LCM.

LCM (12, 16, 24) = 2 2 2 3 2 = 48

Answer: LCM (12, 16, 24) = 48

Special cases of finding NOCs

For example, LCM(60, 15) = 60
Since coprime numbers have no common prime divisors, their least common multiple is equal to the product of these numbers.

On our site, you can also use a special calculator to find the least common multiple online to check your calculations.

If a natural number is only divisible by 1 and itself, then it is called prime.

Any natural number is always divisible by 1 and itself.

The number 2 is the smallest prime number. This is the only even prime number, the rest of the prime numbers are odd.

There are many prime numbers, and the first among them is the number 2. However, there is no last prime number. In the "For Study" section, you can download a table of prime numbers up to 997.

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

• the number 12 is divisible by 1, by 2, by 3, by 4, by 6, by 12;
• 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 evenly divisible (for 12 these are 1, 2, 3, 4, 6 and 12) are called the divisors of the number.

The divisor of a natural number a is such a natural number that divides the given number "a" without a remainder.

A natural number that has more than two factors is called a composite number.

Note that the numbers 12 and 36 have common divisors. These are numbers: 1, 2, 3, 4, 6, 12. The largest divisor of these numbers is 12.

The common divisor of two given numbers "a" and "b" is the number by which both given numbers "a" and "b" are divided without remainder.

Greatest Common Divisor(GCD) of two given numbers "a" and "b" is the largest number by which both numbers "a" and "b" are divisible without a remainder.

Briefly, the greatest common divisor of numbers "a" and "b" is written as follows:

Example: gcd (12; 36) = 12 .

The divisors of numbers in the solution record are denoted by a capital letter "D".

The numbers 7 and 9 have only one common divisor - the number 1. Such numbers are called coprime numbers.

Coprime numbers are natural numbers that have only one common divisor - the number 1. Their GCD is 1.

How to find the greatest common divisor

To find the gcd of two or more natural numbers you need:

• decompose the divisors of numbers into prime factors;

Calculations are conveniently written using a vertical bar. To the left of the line, first write down the dividend, to the right - the divisor. Further in the left column we write down the values ​​of private.

Let's explain right away with an example. Let's factorize the numbers 28 and 64 into prime factors.

Underline the same prime factors in both numbers.
28 = 2 2 7

64 = 2 2 2 2 2 2
We find the product of identical prime factors and write down the answer;
GCD (28; 64) = 2 2 = 4

Answer: GCD (28; 64) = 4

You can arrange the location of the GCD in two ways: in a column (as was done above) or “in a line”.

The first way to write GCD

Find GCD 48 and 36.

GCD (48; 36) = 2 2 3 = 12

The second way to write GCD

Now let's write the GCD search solution in a line. Find GCD 10 and 15.

On our information site, you can also find the greatest common divisor online using the helper program to check your calculations.

Finding the least common multiple, methods, examples of finding the LCM.

The material presented below is a logical continuation of the theory from the article under the heading LCM - Least Common Multiple, definition, examples, relationship between LCM and GCD. Here we will talk about finding the least common multiple (LCM), and pay special attention to solving examples. Let us first show how the LCM of two numbers is calculated in terms of the GCD of these numbers. Next, consider finding the least common multiple by factoring numbers into prime factors. After that, we will focus on finding the LCM of three or more numbers, and also pay attention to the calculation of the LCM of negative numbers.

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Calculation of the least common multiple (LCM) through gcd

One way to find the least common multiple is based on the relationship between LCM and GCD. The existing relationship between LCM and GCD allows you to calculate the least common multiple of two positive integers through the known greatest common divisor. The corresponding formula has the form LCM(a, b)=a b: GCD(a, b). Consider examples of finding the LCM according to the above formula.

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

In this example a=126 , b=70 . Let's use the link of LCM with GCD, which is expressed by the formula LCM(a, b)=a b: GCM(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 according to the written formula.

Find gcd(126, 70) using Euclid's algorithm: 126=70 1+56 , 70=56 1+14 , 56=14 4 , hence gcd(126, 70)=14 .

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

What is LCM(68, 34) ?

Since 68 is evenly divisible by 34 , then gcd(68, 34)=34 . Now we calculate the least common multiple: LCM(68, 34)=68 34:GCD(68, 34)= 68 34:34=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 we make a product of all prime factors of these numbers, after which we exclude from this product all common prime factors that are present in the expansions of these numbers, then the resulting product will be equal to the least common multiple of these numbers.

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

Let's take an example. Let we know that 75=3 5 5 and 210=2 3 5 7 . Compose the product of all factors of these expansions: 2 3 3 5 5 5 7 . Now we exclude from this product all the factors that are present both in the expansion of the number 75 and in the expansion of the number 210 (such 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, LCM(75, 210)= 2 3 5 5 7=1 050 .

After factoring the numbers 441 and 700 into prime factors, find the least common multiple of these numbers.

Let's decompose 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 make a product of all the factors involved in the expansions 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 . So LCM(441, 700)=2 2 3 3 5 5 7 7=44 100 .

LCM(441, 700)= 44 100 .

The rule for finding the LCM using the decomposition of numbers into prime factors can be formulated a little differently. If we add the missing factors from the expansion of the number b 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 all the same numbers 75 and 210, their expansions 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 get the product 2 3 5 5 7 , the value of which is LCM(75, 210) .

Find the least common multiple of 84 and 648.

We first obtain the decomposition 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 get the product 2 2 2 3 3 3 3 7 , which is equal to 4 536 . Thus, the desired least common multiple of the numbers 84 and 648 is 4,536.

Finding the LCM of three or more numbers

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

Let positive integers a 1 , a 2 , …, a k be given, the least common multiple m k of these numbers is found in the sequential calculation m 2 = LCM (a 1 , a 2) , m 3 = LCM (m 2 , a 3) , … , m k =LCM(m k−1 , a k) .

Consider the application of this theorem on the example of finding the least common multiple of four numbers.

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

First we find m 2 = LCM (a 1 , a 2) = LCM (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 , whence LCM(140, 9)=140 9: GCD(140, 9)= 140 9:1=1 260 . That is, m 2 =1 260 .

Now we find m 3 = LCM (m 2 , a 3) = LCM (1 260, 54) . Let's calculate it through gcd(1 260, 54) , which is also determined by the Euclid algorithm: 1 260=54 23+18 , 54=18 3 . Then gcd(1 260, 54)=18 , whence LCM(1 260, 54)= 1 260 54:gcd(1 260, 54)= 1 260 54:18=3 780 . That is, m 3 \u003d 3 780.

It remains to find m 4 = LCM (m 3 , a 4) = LCM (3 780, 250) . To do this, we find GCD(3 780, 250) using the Euclid algorithm: 3 780=250 15+30 , 250=30 8+10 , 30=10 3 . Therefore, gcd(3 780, 250)=10 , hence LCM(3 780, 250)= 3 780 250:gcd(3 780, 250)= 3 780 250:10=94 500 . That is, m 4 \u003d 94 500.

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

LCM(140, 9, 54, 250)=94500 .

In many cases, the least common multiple of three or more numbers is conveniently found using prime factorizations of given numbers. In this case, the following rule should be followed. 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 obtained factors, and so on.

Consider an example of finding the least common multiple using the decomposition of numbers into prime factors.

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

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 expansion of the number 6 does not contain missing factors, since both 2 and 3 are already present in the expansion of the first number 84 . Further 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 is no need to add factors 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 .

Therefore, LCM(84, 6, 48, 7, 143)=48048 .

LCM(84, 6, 48, 7, 143)=48048 .

Finding the Least Common Multiple of Negative Numbers

Sometimes there are tasks in which you need to find the least common multiple of numbers, among which one, several or all numbers are negative. In these cases, all negative numbers must be replaced by their opposite numbers, after which the LCM of positive numbers should be found. This is the way to find the LCM of negative numbers. For example, LCM(54, −34)=LCM(54, 34) and LCM(−622, −46, −54, −888)= LCM(622, 46, 54, 888) .

We can do this because the set of multiples of a is the same as the set of multiples of −a (a and −a are opposite numbers). Indeed, let b be some multiple of a , then b is divisible by a , and the concept of divisibility asserts the existence of such an integer q that b=a q . But the equality b=(−a)·(−q) will also be true, which, by virtue of the same concept of divisibility, means that b is divisible by −a , that is, b is a multiple of −a . The converse statement is also true: if b is some multiple of −a , then b is also a multiple of a .

Find the least common multiple of the negative numbers −145 and −45.

Let's replace the negative numbers −145 and −45 with their opposite numbers 145 and 45 . We have LCM(−145, −45)=LCM(145, 45) . Having determined gcd(145, 45)=5 (for example, using the Euclid algorithm), we calculate LCM(145, 45)=145 45:gcd(145, 45)= 145 45:5=1 305 . Thus, the least common multiple of the negative integers −145 and −45 is 1,305 .

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We continue to study division. In this lesson, we will look at concepts such as GCD and NOC.

GCD is the greatest common divisor.

NOC is the least common multiple.

The topic is rather boring, but it is necessary to understand it. Without understanding this topic, you will not be able to work effectively with fractions, which are a real obstacle in mathematics.

Greatest Common Divisor

Definition. Greatest Common Divisor of Numbers a and b a and b divided without remainder.

In order to understand this definition well, we substitute instead of variables a and b any two numbers, for example, instead of a variable a substitute the number 12, and instead of the variable b number 9. Now let's try to read this definition:

Greatest Common Divisor of Numbers 12 and 9 is the largest number by which 12 and 9 divided without remainder.

It is clear from the definition that we are talking about a common divisor of the numbers 12 and 9, and this divisor is the largest of all existing divisors. This greatest common divisor (gcd) must be found.

To find the greatest common divisor of two numbers, three methods are used. The first method is quite time-consuming, but it allows you to understand the essence of the topic well and feel its whole meaning.

The second and third methods are quite simple and make it possible to quickly find the GCD. We will consider all three methods. And what to apply in practice - you choose.

The first way is to find all possible divisors of two numbers and choose the largest of them. Let's consider this method in the following example: find the greatest common divisor of the numbers 12 and 9.

First, we find all possible divisors of the number 12. To do this, we divide 12 into all divisors in the range from 1 to 12. If the divisor allows us to divide 12 without a remainder, then we will highlight it in blue and make an appropriate explanation in brackets.

12: 1 = 12
(12 divided by 1 without a remainder, so 1 is a divisor of 12)

12: 2 = 6
(12 divided by 2 without a remainder, so 2 is a divisor of 12)

12: 3 = 4
(12 divided by 3 without a remainder, so 3 is a divisor of 12)

12: 4 = 3
(12 divided by 4 without a remainder, so 4 is a divisor of 12)

12:5 = 2 (2 left)
(12 is not divided by 5 without a remainder, so 5 is not a divisor of 12)

12: 6 = 2
(12 divided by 6 without a remainder, so 6 is a divisor of 12)

12: 7 = 1 (5 left)
(12 is not divided by 7 without a remainder, so 7 is not a divisor of 12)

12: 8 = 1 (4 left)
(12 is not divided by 8 without a remainder, so 8 is not a divisor of 12)

12:9 = 1 (3 left)
(12 is not divided by 9 without a remainder, so 9 is not a divisor of 12)

12: 10 = 1 (2 left)
(12 is not divided by 10 without a remainder, so 10 is not a divisor of 12)

12:11 = 1 (1 left)
(12 is not divided by 11 without a remainder, so 11 is not a divisor of 12)

12: 12 = 1
(12 divided by 12 without a remainder, so 12 is a divisor of 12)

Now let's find the divisors of the number 9. To do this, check all the divisors from 1 to 9

9: 1 = 9
(9 divided by 1 without a remainder, so 1 is a divisor of 9)

9: 2 = 4 (1 left)
(9 is not divided by 2 without a remainder, so 2 is not a divisor of 9)

9: 3 = 3
(9 divided by 3 without a remainder, so 3 is a divisor of 9)

9: 4 = 2 (1 left)
(9 is not divided by 4 without a remainder, so 4 is not a divisor of 9)

9:5 = 1 (4 left)
(9 is not divided by 5 without a remainder, so 5 is not a divisor of 9)

9: 6 = 1 (3 left)
(9 did not divide by 6 without a remainder, so 6 is not a divisor of 9)

9:7 = 1 (2 left)
(9 is not divided by 7 without a remainder, so 7 is not a divisor of 9)

9:8 = 1 (1 left)
(9 is not divided by 8 without a remainder, so 8 is not a divisor of 9)

9: 9 = 1
(9 divided by 9 without a remainder, so 9 is a divisor of 9)

Now write down the divisors of both numbers. The numbers highlighted in blue are the divisors. Let's write them out:

Having written out the divisors, you can immediately determine which one is the largest and most common.

By definition, the greatest common divisor of 12 and 9 is the number by which 12 and 9 are evenly divisible. The greatest and common divisor of the numbers 12 and 9 is the number 3

Both the number 12 and the number 9 are divisible by 3 without a remainder:

So gcd (12 and 9) = 3

The second way to find GCD

Now consider the second way to find the greatest common divisor. The essence of this method is to decompose both numbers into prime factors and multiply the common ones.

Example 1. Find GCD of numbers 24 and 18

First, let's factor both numbers into prime factors:

Now we multiply their common factors. In order not to get confused, the common factors can be underlined.

We look at the decomposition of the number 24. Its first factor is 2. We are looking for the same factor in the decomposition of the number 18 and see that it is also there. We underline both twos:

Again we look at the decomposition of the number 24. Its second factor is also 2. We are looking for the same factor in the decomposition of the number 18 and see that it is not there for the second time. Then we don't highlight anything.

The next two in the expansion of the number 24 is also missing in the expansion of the number 18.

We pass to the last factor in the decomposition of the number 24. This is the factor 3. We are looking for the same factor in the decomposition of the number 18 and see that it is also there. We emphasize both threes:

So, the common factors of the numbers 24 and 18 are the factors 2 and 3. To get the GCD, these factors must be multiplied:

So gcd (24 and 18) = 6

The third way to find GCD

Now consider the third way to find the greatest common divisor. The essence of this method lies in the fact that the numbers to be searched for the greatest common divisor are decomposed into prime factors. Then, from the decomposition of the first number, factors that are not included in the decomposition of the second number are deleted. The remaining numbers in the first expansion are multiplied and get GCD.

For example, let's find the GCD for the numbers 28 and 16 in this way. First of all, we decompose these numbers into prime factors:

We got two expansions: and

Now, from the expansion of the first number, we delete the factors that are not included in the expansion of the second number. The expansion of the second number does not include seven. We will delete it from the first expansion:

Now we multiply the remaining factors and get the GCD:

The number 4 is the greatest common divisor of the numbers 28 and 16. Both of these numbers are divisible by 4 without a remainder:

Example 2 Find GCD of numbers 100 and 40

Factoring out the number 100

Factoring out the number 40

We got two expansions:

Now, from the expansion of the first number, we delete the factors that are not included in the expansion of the second number. The expansion of the second number does not include one five (there is only one five). We delete it from the first decomposition

Multiply the remaining numbers:

We got the answer 20. So the number 20 is the greatest common divisor of the numbers 100 and 40. These two numbers are divisible by 20 without a remainder:

GCD (100 and 40) = 20.

Example 3 Find the gcd of the numbers 72 and 128

Factoring out the number 72

Factoring out the number 128

2×2×2×2×2×2×2

Now, from the expansion of the first number, we delete the factors that are not included in the expansion of the second number. The expansion of the second number does not include two triplets (there are none at all). We delete them from the first expansion:

We got the answer 8. So the number 8 is the greatest common divisor of the numbers 72 and 128. These two numbers are divisible by 8 without a remainder:

GCD (72 and 128) = 8

Finding GCD for Multiple Numbers

The greatest common divisor can be found for several numbers, and not just for two. For this, the numbers to be searched for the greatest common divisor are decomposed into prime factors, then the product of the common prime factors of these numbers is found.

For example, let's find the GCD for the numbers 18, 24 and 36

Factoring the number 18

Factoring the number 24

Factoring the number 36

We got three expansions:

Now we select and underline the common factors in these numbers. Common factors must be included in all three numbers:

We see that the common factors for the numbers 18, 24 and 36 are factors 2 and 3. By multiplying these factors, we get the GCD we are looking for:

We got the answer 6. So the number 6 is the greatest common divisor of the numbers 18, 24 and 36. These three numbers are divisible by 6 without a remainder:

GCD (18, 24 and 36) = 6

Example 2 Find gcd for numbers 12, 24, 36 and 42

Let's factorize each number. Then we find the product of the common factors of these numbers.

Factoring the number 12

Factoring the number 42

We got four expansions:

Now we select and underline the common factors in these numbers. Common factors must be included in all four numbers:

We see that the common factors for the numbers 12, 24, 36, and 42 are factors 2 and 3. By multiplying these factors, we get the GCD we are looking for:

We got the answer 6. So the number 6 is the greatest common divisor of the numbers 12, 24, 36 and 42. These numbers are divisible by 6 without a remainder:

gcd(12, 24, 36 and 42) = 6

From the previous lesson, we know that if some number is divided by another without a remainder, it is called a multiple of this number.

It turns out that a multiple can be common to several numbers. And now we will be interested in a multiple of two numbers, while it should be as small as possible.

Definition. Least common multiple (LCM) of numbers a and b- a and b a and number b.

Definition contains two variables a and b. Let's substitute any two numbers for these variables. For example, instead of a variable a substitute the number 9, and instead of the variable b let's substitute the number 12. Now let's try to read the definition:

Least common multiple (LCM) of numbers 9 and 12 - is the smallest number that is a multiple of 9 and 12 . In other words, it is such a small number that is divisible without a remainder by the number 9 and on the number 12 .

It is clear from the definition that the LCM is the smallest number that is divisible without a remainder by 9 and 12. This LCM is required to be found.

There are two ways to find the least common multiple (LCM). The first way is that you can write down the first multiples of two numbers, and then choose among these multiples such a number that will be common to both numbers and small. Let's apply this method.

First of all, let's find the first multiples for the number 9. To find multiples for 9, you need to multiply this nine by the numbers from 1 to 9 in turn. The answers you get will be multiples of the number 9. So, let's start. Multiples will be highlighted in red:

Now we find multiples for the number 12. To do this, we multiply 12 by all the numbers 1 to 12 in turn.

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 consider 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.

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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.

Solution

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.

Solution

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 becomes 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.

Solution

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.

Solution

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 sequentially 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 .

Solution

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;
• to the product obtained at the previous stage, we add the missing factors of the third number, 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 .

Solution

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 .

Solution

Let's change the numbers − 145 and − 45 to their opposite numbers 145 and 45 . Now, according to 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 .

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