The least common multiple of two numbers is directly related to the greatest common divisor of those numbers. This connection between GCD and NOC is determined 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 M is some multiple of the 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 also 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 relatively prime numbers. Consequently, the condition obtained in the previous paragraph that a · k is divisible by b can be reformulated as follows: a 1 · d · k is divided by b 1 · d , and this, due to the properties of divisibility, is equivalent to the condition that a 1 · k is divisible by b 1.

You also need to write down two important corollaries from the theorem considered.

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

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

Least common multiple of coprime positive numbers a and b are equal to their product.

The rationale for this fact is quite obvious. Since a and b are relatively prime, then gcd(a, b)=1, therefore, GCD(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 the common multiples of the numbers m k-1 and a k , therefore, coincide with the common multiples of the number m k . And since the smallest positive multiple of the number m k is the number m k itself, then the smallest common multiple of the numbers a 1, a 2, ..., a k is m k.

References.

• Vilenkin N.Ya. and others. Mathematics. 6th grade: textbook for general education institutions.
• Vinogradov I.M. Fundamentals of number theory.
• Mikhelovich Sh.H. Number theory.
• Kulikov L.Ya. and others. Collection of problems in algebra and number theory: Tutorial for students of physics and mathematics. specialties of pedagogical institutes.

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 times 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 required 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.

Let's look at three ways to find the least common multiple.

Finding by factorization

The first method is to find the least common multiple by factoring the given numbers into prime factors.

Let's say we need to find the LCM of the numbers: 99, 30 and 28. To do this, let's factor each of these numbers into prime factors:

For the desired number to be divisible by 99, 30 and 28, it is necessary and sufficient that it includes all the prime factors of these divisors. To do this, we need to take all the prime factors of these numbers to the greatest possible degree and multiply them together:

2 2 3 2 5 7 11 = 13,860

Thus, LCM (99, 30, 28) = 13,860. No other number less than 13,860 is divisible by 99, 30, or 28.

To find the least common multiple of given numbers, you factor them into their prime factors, then take each prime factor with the largest exponent it appears in, and multiply those factors together.

Since relatively prime numbers do not have common prime factors, their least common multiple is equal to the product of these numbers. For example, three numbers: 20, 49 and 33 are relatively prime. That's why

LCM (20, 49, 33) = 20 49 33 = 32,340.

The same must be done when finding the least common multiple of different prime numbers. For example, LCM (3, 7, 11) = 3 7 11 = 231.

Finding by selection

The second method is to find the least common multiple by selection.

Example 1. When the largest of given numbers is divided by another given number, then the LCM of these numbers is equal to the largest of them. For example, given four numbers: 60, 30, 10 and 6. Each of them is divisible by 60, therefore:

LCM(60, 30, 10, 6) = 60

In other cases, to find the least common multiple, the following procedure is used:

1. Determine the largest number from the given numbers.
2. Next, we find the numbers that are multiples of the largest number by multiplying it by natural numbers in ascending order and checking whether the remaining numbers are divisible by the resulting product.

Example 2. Given three numbers 24, 3 and 18. We determine the largest of them - this is the number 24. Next, we find the numbers that are multiples of 24, checking whether each of them is divisible by 18 and 3:

24 · 1 = 24 - divisible by 3, but not divisible by 18.

24 · 2 = 48 - divisible by 3, but not divisible by 18.

24 · 3 = 72 - divisible by 3 and 18.

Thus, LCM (24, 3, 18) = 72.

Finding by sequentially finding the LCM

The third method is to find the least common multiple by sequentially finding the LCM.

The LCM of two given numbers is equal to the product of these numbers divided by their greatest common divisor.

Example 1. Find the LCM of two given numbers: 12 and 8. Determine their greatest common divisor: GCD (12, 8) = 4. Multiply these numbers:

We divide the product by their gcd:

Thus, LCM (12, 8) = 24.

To find the LCM of three or more numbers, use the following procedure:

1. First, find the LCM of any two of these numbers.
2. Then, LCM of the found least common multiple and the third given number.
3. Then, the LCM of the resulting least common multiple and the fourth number, etc.
4. Thus, the search for LCM continues as long as there are numbers.

Example 2. Let's find the LCM of three given numbers: 12, 8 and 9. We already found the LCM of the numbers 12 and 8 in the previous example (this is the number 24). It remains to find the least common multiple of the number 24 and the third given number - 9. Determine their greatest common divisor: GCD (24, 9) = 3. Multiply the LCM with the number 9:

We divide the product by their gcd:

Thus, LCM (12, 8, 9) = 72.

The topic “Multiples” is studied in grade 5 secondary school. Its goal is to improve written and oral mathematical calculation skills. In this lesson, new concepts are introduced - “multiple numbers” and “divisors”, the technique of finding divisors and multiples of a natural number, and the ability to find LCM in various ways are practiced.

This topic is very important. Knowledge of 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.

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

You need to prove that the number 125 is a multiple of the number 5. To do this, you need to 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 LOC.

1. If you need to find a common multiple of 2 numbers (for example, 80 and 20), where one of them (80) is divisible by the other (20), then this number (80) is the least 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 look at the last example. 6 and 7 in relation to 42 are divisors. They divide a multiple of a number without a remainder.

In this example, 6 and 7 are paired factors. Their product is equal to the most multiple 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.

Another example involves determining whether 9 is a divisor of 42.

42:9=4 (remainder 6)

Answer: 9 is not a divisor of 42 because the answer has a remainder.

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

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

Namely: gcd (a, b) x gcd (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 factor these numbers into prime factors and write them as a product of powers:

168=2³x3¹x7¹

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, 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 accepted designation short name, 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 No. 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 received numbers must participate in the calculation of the final result. For each multiplier, the most large number occurrences. 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 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 reduce fractions to a common denominator. What is usually studied in grades 5-6 high school. It is also additionally a common divisor for all multiples, if such conditions are present in the problem. Such an expression can find multiples not only of two numbers, but also of much more more- 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 display a 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 involved in this calculation there is also a greatest common divisor, which is calculated according to similar principles and is often encountered in neighboring problems. The difference is small, but quite significant, LCM involves calculating a number that is divided by all given initial values, and GCD involves calculating highest value by which the original numbers are divided.