How to find the least common multiple of two numbers. Finding the least common multiple: methods, examples of finding the LCM

The online calculator allows you to quickly find the greatest common factor and least common multiple for both two and for any other number of numbers.

Calculator for finding GCD and LCM

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Found GCD and NOC: 5806

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  • Enter numbers in the input field
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  • click the button "Find GCD and LCM"

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  • Numbers are entered separated by a space, period or comma
  • The length of the entered numbers is not limited, so finding the GCD and LCM of long numbers will not be difficult

What are GCD and NOC?

Greatest common divisor multiple numbers is the largest natural integer by which all original numbers are divisible without a remainder. The greatest common factor is abbreviated as Gcd.
Least common multiple multiple numbers is the smallest number that is divisible by each of the original numbers without a remainder. The least common multiple is abbreviated as NOC.

How to check that a number is divisible by another number without a remainder?

To find out whether one number is divisible by another without a remainder, you can use some of the divisibility properties of numbers. Then, by combining them, one can check divisibility into some of them and their combinations.

Some signs of divisibility of numbers

1. The criterion for divisibility of a number by 2
To determine whether a number is divisible by two (whether it is even), it is enough to look at the last digit of this number: if it is 0, 2, 4, 6 or 8, then the number is even, which means it is divisible by 2.
Example: determine if 34938 is divisible by 2.
Solution: look at the last digit: 8 - so the number is divisible by two.

2. The sign of divisibility of a number by 3
A number is divisible by 3 when the sum of its digits is divisible by three. Thus, to determine if a number is divisible by 3, you need to calculate the sum of the digits and check if it is divisible by 3. Even if the sum of the digits is very large, you can repeat the same process again.
Example: determine if 34938 is divisible by 3.
Solution: we count the sum of digits: 3 + 4 + 9 + 3 + 8 = 27.27 is divisible by 3, which means that the number is divisible by three.

3. The sign of divisibility of a number by 5
A number is divisible by 5 when its last digit is zero or five.
Example: determine if 34938 is divisible by 5.
Solution: look at the last digit: 8 - means the number is NOT divisible by five.

4. The sign of divisibility of a number by 9
This feature is very similar to the divisibility by three: a number is divisible by 9 when the sum of its digits is divisible by 9.
Example: determine if 34938 is divisible by 9.
Solution: we count the sum of the digits: 3 + 4 + 9 + 3 + 8 = 27.27 is divisible by 9, which means that the number is divisible by nine.

How to find gcd and LCM of two numbers

How to find the gcd of two numbers

The simplest way to calculate the greatest common divisor of two numbers is to find all possible divisors of those numbers and choose the largest one.

Let us consider this method using the example of finding the GCD (28, 36):

  1. Factor both numbers: 28 = 1 2 2 7, 36 = 1 2 2 3 3
  2. We find the common factors, that is, those that both numbers have: 1, 2 and 2.
  3. We calculate the product of these factors: 1 · 2 · 2 = 4 - this is the greatest common divisor of the numbers 28 and 36.

How to find the LCM of two numbers

There are two most common ways to find the least multiple of two numbers. The first way is that you can write out the first multiples of two numbers, and then choose among them such a number that will be common to both numbers and at the same time the smallest. And the second is to find the GCD of these numbers. Let's consider only it.

To calculate the LCM, you need to calculate the product of the original numbers and then divide it by the previously found GCD. Find the LCM for the same numbers 28 and 36:

  1. Find the product of the numbers 28 and 36: 28 36 = 1008
  2. GCD (28, 36), as is already known, is equal to 4
  3. LCM (28, 36) = 1008/4 = 252.

Finding GCD and LCM for several numbers

The greatest common factor can be found for several numbers, not just two. For this, the numbers to be searched for the greatest common factor are decomposed into prime factors, then the product of the common prime factors of these numbers is found. Also, to find the GCD of several numbers, you can use the following relationship: Gcd (a, b, c) = gcd (gcd (a, b), c).

A similar relationship applies to the least common multiple: LCM (a, b, c) = LCM (LCM (a, b), c)

Example: find GCD and LCM for numbers 12, 32 and 36.

  1. First, factor the numbers: 12 = 1 2 2 3, 32 = 1 2 2 2 2 2 2, 36 = 1 2 2 3 3 3.
  2. Let's find the common factors: 1, 2 and 2.
  3. Their product will give GCD: 1 2 2 = 4
  4. Let us now find the LCM: for this we first find the LCM (12, 32): 12 · 32/4 = 96.
  5. To find the LCM of all three numbers, you need to find the GCD (96, 36): 96 = 1 2 2 2 2 2 2 3, 36 = 1 2 2 3 3, GCD = 1 2 2 3 = 12.
  6. LCM (12, 32, 36) = 96 36/12 = 288.

The least common multiple of two numbers is directly related to the greatest common divisor of those numbers. This the relationship between the GCD and the NOC is defined by the following theorem.

Theorem.

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

Proof.

Let be M - any multiple of numbers a and b. That is, M is divisible by a, and by the definition of divisibility there is some integer k such that the equality M = a · k is true. But M is divisible by b, then a · k is divisible by b.

Let's denote gcd (a, b) as d. Then we can write the equalities a = a 1 d and b = b 1 d, and a 1 = a: d and b 1 = b: d will be coprime numbers. Therefore, the condition obtained in the previous paragraph that a k is divisible by b can be reformulated as follows: a 1 d k is divisible by b 1 d, and this, due to the divisibility properties, is equivalent to the condition that a 1 k is divisible by b 1 .

You also need to write down two important consequences of the considered theorem.

    Common multiples of two numbers are the same as multiples of their least common multiple.

    This is indeed so, since any common multiple M of the numbers a and b is determined by the equality M = LCM (a, b) t for some integer value of t.

    The least common multiple of coprime positive numbers a and b is equal to their product.

    The rationale for this fact is fairly obvious. 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

Finding the least common multiple of three or more numbers can be reduced to sequentially finding the LCM of two numbers. How this is done is indicated in the following theorem. A 1, a 2,…, a k coincide with common multiples of m k-1 and a k, therefore, coincide with multiples of m k. And since the smallest positive multiple of the number m k is the number m k itself, the least common multiple of the numbers a 1, a 2,…, a k is m k.

Bibliography.

  • Vilenkin N.Ya. and other Mathematics. Grade 6: textbook for educational institutions.
  • Vinogradov I.M. Fundamentals of number theory.
  • Mikhelovich Sh.Kh. Number theory.
  • Kulikov L.Ya. and others. Collection of problems in algebra and number theory: a textbook for students of physics and mathematics. specialties of pedagogical institutes.

The topic "Multiples" is studied in the 5th grade of a comprehensive school. Its goal is to improve the written and oral skills of mathematical calculations. In this lesson, new concepts are introduced - "multiples" and "divisors", the technique of finding divisors and multiples of a natural number, the ability to find LCM in various ways is being worked out.

This topic is very important. Knowledge on it can be applied when solving examples with fractions. To do this, you need to find the common denominator by calculating the least common multiple (LCM).

A multiple of A is an integer that is divisible by A without a remainder.

Each natural number has an infinite number of multiples of it. It itself is considered the smallest. The multiple cannot be less than the number itself.

We need to prove that 125 is a multiple of 5. To do this, divide the first number by the second. If 125 is divisible by 5 without a remainder, then the answer is yes.

This method is applicable for small numbers.

There are special cases when calculating the LCM.

1. If you need to find a common multiple for 2 numbers (for example, 80 and 20), where one of them (80) is divided without a remainder by the other (20), then this number (80) is the smallest multiple of these two numbers.

LCM (80, 20) = 80.

2. If two do not have a common divisor, then we can say that their LCM is the product of these two numbers.

LCM (6, 7) = 42.

Let's take a look at the last example. 6 and 7 with respect to 42 are divisors. They divide a multiple without a remainder.

In this example, 6 and 7 are paired divisors. Their product is equal to the most multiple of the number (42).

A number is called prime if it is divisible only by itself or by 1 (3: 1 = 3; 3: 3 = 1). The rest are called composite.

In another example, you need to determine if 9 is a divisor of 42.

42: 9 = 4 (remainder 6)

Answer: 9 is not a divisor of 42, because there is a remainder in the answer.

The divisor differs from the multiple in that the divisor is the number by which the natural numbers are divided, and the multiple itself is divisible by this number.

Greatest common divisor of numbers a and b, multiplied by their smallest multiple, will give the product of the numbers themselves a and b.

Namely: GCD (a, b) x LCM (a, b) = a x b.

Common multiples for more complex numbers are found in the following way.

For example, find the LCM for 168, 180, 3024.

We decompose these numbers into prime factors, write them in the form of a product of degrees:

168 = 2³х3¹х7¹

2⁴х3³х5¹х7¹ = 15120

LCM (168, 180, 3024) = 15120.

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

Definition

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 adopted for designation, assembled from the first letters.

Ways to get the number

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

Example No. 1

For the simplest example, schools usually use simple, single, or two-digit numbers. For example, you need to solve the following problem, 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 variant of the task is much more difficult. Given the numbers 300 and 1260, 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 already received data. Each of the numbers obtained must participate in the calculation of the final result. For each factor, the largest number of occurrences is taken from the original numbers. The LCM is the total number, therefore, the factors of the numbers must be repeated in it all to one, even those that are present in one copy. Both original 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 the powers presented in the equation. All that remains is 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) LCM = 6300.

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

Examination:

6300/300 = 21 - true;

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 LCM mean in mathematics

As you know, in mathematics there is not a single useless function, this is no exception. The most common use for 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. A similar expression can find a multiple not only of two numbers, but also to a much larger number - three, five, and so on. The more numbers - the more actions in the task, but the complexity does not increase from this.

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 - in this example, the factorization is described in detail, without cancellation.

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 the 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 complete simplification, if possible, expanding to the level of unambiguous ones.

Examination:

1) 3000/250 = 12 - true;

2) 3000/600 = 5 - true;

3) 3000/1500 = 2 - true.

This method does not require any gimmicks or genius-level abilities, everything is simple and straightforward.

Another way

In mathematics, a lot is connected, a lot can be solved in two or more ways, the same applies to 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 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, from 1 to infinity, are written in a row, sometimes 3-5 points are enough, the second and subsequent numbers are subjected to the same computational process. Everything happens until the 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 LCM. Among the processes associated with this computation, 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, the LCM assumes the calculation of a number that is divided by all the given initial values, and the GCD assumes the calculation of the largest value by which the original numbers are divided.

Schoolchildren are given a lot of math assignments. Among them, tasks with the following formulation are very common: there are two meanings. How do I 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 with different denominators. In this 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 decide on the term multiple... Most often, the formulation of this concept sounds as follows: a multiple of a certain value of A is called a natural number that will be divisible by A. So, for 4, the multiples will be 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 lowest value for certain indicators, let's move on to how to find it.

Find the LCM

The smallest multiple of two or more exponents is the smallest natural number that is fully divisible by all specified numbers.

There are several ways to find such a value., consider the following methods:

  1. If the numbers are small, then write down all divisible by it in a line. 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 it is large or you need to find a multiple of 3 or more values, then another technique should be used, which involves the decomposition of numbers into prime factors. First, lay out the largest of the indicated, then all the rest. Each of them has its own number of factors. As an example, let's expand 20 (2 * 2 * 5) and 50 (5 * 5 * 2). For the smaller one, underline the factors and add to the largest. The result is 100, which will be the smallest 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 us expand each of them: 16 = 2 * 2 * 2 * 2, 24 = 2 * 2 * 2 * 3, 36 = 2 * 2 * 3 * 3. Only two twos from the expansion of 16 were not included in the expansion of the largest. Add them and get 144, which is the smallest result for the previously indicated numerical values.

We now 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 look for an NOC, if the previous ones do not help.

How to find GCD and LCM.

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 divided into others without a remainder, then the lowest multiple of these numbers is equal to it (LCM 60 and 15 is 15);
  • coprime numbers have no 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 the cases of decomposition of composite numbers, which are the topic of individual articles and even candidate dissertations.

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

Few examples

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

  1. Find the LCM (35; 40). We expand first 35 = 5 * 7, then 40 = 5 * 8. Add 8 to the smallest number and get the LCM 280.
  2. LCM (45; 54). We lay out each of them: 45 = 3 * 3 * 5 and 54 = 3 * 3 * 6. Add to 45 the number 6. We get the LCM equal to 270.
  3. Well, the last example. There are 5 and 4. There are no prime 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 LCM is located, what are the nuances and what is the meaning of such manipulations.

Finding an NOC is much easier than it might initially seem. For this, both simple decomposition and multiplication of simple values ​​by each other are used.... The ability to work with this branch of mathematics helps in the further study of mathematical topics, especially fractions of varying degrees of complexity.

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

Video

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