What is a nok of numbers. Least common multiple of LCM

Schoolchildren are given a lot of tasks in mathematics. Among them, very often there are problems with the following formulation: there are two meanings. How to find the least common multiple of given numbers? It is necessary to be able to perform such tasks, since the acquired skills are used to work with fractions when different denominators. In this article we will look at how to find LOC and basic concepts.

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

Moreover, the number of divisors for a specific value can be limited, but the multiples are infinitely many. There is also the same value for natural values. This is an indicator that is divided into them without a remainder. Having understood the concept of the smallest value for certain indicators, let's move on to how to find it.

Finding the NOC

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

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

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

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

How to find GCD and NOC.

Private methods of finding

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

if one of the numbers is divisible by the others without a remainder, then the lowest multiple of these numbers is equal to it (the LCM of 60 and 15 is 15);
relatively prime numbers have no common prime factors. Their smallest value is equal to the product of these numbers. Thus, for the numbers 7 and 8 it will be 56;
the same rule works for other cases, including special ones, which can be read about in specialized literature. This should also include cases of decomposition of composite numbers, which are the 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 unequal denominators.

Few examples

Let's look at a few examples that will help you understand the principle of finding the least multiple:

Find the LOC (35; 40). We first decompose 35 = 5*7, then 40 = 5*8. Add 8 to the smallest number and get LOC 280.
NOC (45; 54). We decompose each of them: 45 = 3*3*5 and 54 = 3*3*6. We add the number 6 to 45. We get an LCM equal to 270.
Well, the last example. There are 5 and 4. There are no prime multiples of them, so the least common multiple in this case will be their product, which is equal to 20.

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

Finding NOC is much easier than it might initially seem. To do this, both simple expansion and multiplication are used simple values on top of each other. The ability to work with this section of mathematics helps with further study of mathematical topics, especially fractions varying degrees complexity.

Don't forget to solve examples periodically various methods, this develops the logical apparatus and allows you to remember numerous terms. Learn how to find such an exponent and you will be able to do well in the rest of the math sections. Happy learning math!

Video

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

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 simple factors and write them as a product of powers:

168=2³x3¹x7¹

2⁴х3³х5¹х7¹=15120

LCM(168, 180, 3024) = 15120.

A multiple is a number that is divisible by a given number without a remainder. The least common multiple (LCM) of a group of numbers is the smallest number that is divisible by each number in the group without leaving a remainder. To find the least common multiple, you need to find the prime factors of given numbers. The LCM can also be calculated using a number of other methods that apply to groups of two or more numbers.

Steps

Series of multiples

Look at these numbers. The method described here is best used when given two numbers, each of which is less than 10. If larger numbers are given, use a different method.

For example, find the least common multiple of 5 and 8. These are small numbers, so you can use this method.

A multiple is a number that is divisible by a given number without a remainder. Multiples can be found in the multiplication table.
- For example, numbers that are multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40.
Write down a series of numbers that are multiples of the first number. Do this under multiples of the first number to compare two sets of numbers.
- For example, numbers that are multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, and 64.
Find the smallest number that is present in both sets of multiples. You may have to write long rows multiples to find total number. The smallest number that is present in both sets of multiples is the least common multiple.
- For example, the smallest number, which is present in the series of multiples of 5 and 8, is the number 40. Therefore, 40 is the least common multiple of 5 and 8.
Prime factorization
1. Look at these numbers. The method described here is best used when given two numbers, each of which is greater than 10. If smaller numbers are given, use a different method.
  - For example, find the least common multiple of the numbers 20 and 84. Each of the numbers is greater than 10, so you can use this method.
2. Factor the first number into prime factors. That is, you need to find such prime numbers that, when multiplied, will yield a given number. Once you have found the prime factors, write them as equalities.
  - For example, 2 × 10 = 20 (\displaystyle (\mathbf (2) )\times 10=20) And 2 × 5 = 10 (\displaystyle (\mathbf (2) )\times (\mathbf (5) )=10). Thus, the prime factors of the number 20 are the numbers 2, 2 and 5. Write them as an expression: .
3. Factor the second number into prime factors. Do this in the same way as you factored the first number, that is, find such prime numbers that, when multiplied, will yield the given number.
  - For example, 2 × 42 = 84 (\displaystyle (\mathbf (2) )\times 42=84), 7 × 6 = 42 (\displaystyle (\mathbf (7) )\times 6=42) And 3 × 2 = 6 (\displaystyle (\mathbf (3) )\times (\mathbf (2) )=6). Thus, the prime factors of the number 84 are the numbers 2, 7, 3 and 2. Write them as an expression: .
4. Write down the factors common to both numbers. Write such factors as a multiplication operation. As you write each factor, cross it out in both expressions (expressions that describe factorizations of numbers into prime factors).
  - For example, both numbers have a common factor of 2, so write 2 × (\displaystyle 2\times ) and cross out the 2 in both expressions.
  - What both numbers have in common is another factor of 2, so write 2 × 2 (\displaystyle 2\times 2) and cross out the second 2 in both expressions.
5. Add the remaining factors to the multiplication operation. These are factors that are not crossed out in both expressions, that is, factors that are not common to both numbers.
  - For example, in the expression 20 = 2 × 2 × 5 (\displaystyle 20=2\times 2\times 5) Both twos (2) are crossed out because they are common factors. The factor 5 is not crossed out, so write the multiplication operation like this: 2 × 2 × 5 (\displaystyle 2\times 2\times 5)
  - In expression 84 = 2 × 7 × 3 × 2 (\displaystyle 84=2\times 7\times 3\times 2) both twos (2) are also crossed out. The factors 7 and 3 are not crossed out, so write the multiplication operation like this: 2 × 2 × 5 × 7 × 3 (\displaystyle 2\times 2\times 5\times 7\times 3).
6. Calculate the least common multiple. To do this, multiply the numbers in the written multiplication operation.
  - For example, 2 × 2 × 5 × 7 × 3 = 420 (\displaystyle 2\times 2\times 5\times 7\times 3=420). So the least common multiple of 20 and 84 is 420.
Finding common factors
1. Draw a grid like for a game of tic-tac-toe. Such a grid consists of two parallel lines that intersect (at right angles) with another two parallel lines. This will give you three rows and three columns (the grid looks a lot like the # icon). Write the first number in the first line and second column. Write the second number in the first row and third column.
  - For example, find the least common multiple of the numbers 18 and 30. Write the number 18 in the first row and second column, and write the number 30 in the first row and third column.
2. Find the divisor common to both numbers. Write it down in the first row and first column. It is better to look for prime factors, but this is not a requirement.
  - For example, 18 and 30 are even numbers, so their common divisor the number will be 2. So write 2 in the first row and first column.
3. Divide each number by the first divisor. Write each quotient under the appropriate number. A quotient is the result of dividing two numbers.
  - For example, 18 ÷ 2 = 9 (\displaystyle 18\div 2=9), so write 9 under 18.
  - 30 ÷ 2 = 15 (\displaystyle 30\div 2=15), so write down 15 under 30.
4. Find the divisor common to both quotients. If there is no such divisor, skip two next steps. Otherwise, write the divisor in the second row and first column.
  - For example, 9 and 15 are divisible by 3, so write 3 in the second row and first column.
5. Divide each quotient by its second divisor. Write each division result under the corresponding quotient.
  - For example, 9 ÷ 3 = 3 (\displaystyle 9\div 3=3), so write 3 under 9.
  - 15 ÷ 3 = 5 (\displaystyle 15\div 3=5), so write 5 under 15.
6. If necessary, add additional cells to the grid. Repeat the described steps until the quotients have a common divisor.
7. Circle the numbers in the first column and last row of the grid. Then write the selected numbers as a multiplication operation.
  - For example, the numbers 2 and 3 are in the first column, and the numbers 3 and 5 are in the last row, so write the multiplication operation like this: 2 × 3 × 3 × 5 (\displaystyle 2\times 3\times 3\times 5).
8. Find the result of multiplying numbers. This will calculate the least common multiple of two given numbers.
  - For example, 2 × 3 × 3 × 5 = 90 (\displaystyle 2\times 3\times 3\times 5=90). So the least common multiple of 18 and 30 is 90.
Euclid's algorithm
1. Remember the terminology associated with the division operation. The dividend is the number that is being divided. The divisor is the number that is being divided by. A quotient is the result of dividing two numbers. A remainder is the number left when two numbers are divided.
  - For example, in the expression 15 ÷ 6 = 2 (\displaystyle 15\div 6=2) ost. 3:
    15 is the dividend
    6 is a divisor
    2 is quotient
    3 is the remainder.

To understand how to calculate the LCM, you must first determine the meaning of the term “multiple.”

A multiple of A is a natural number that is divisible by A without a remainder. Thus, numbers that are multiples of 5 can be considered 15, 20, 25, and so on.

There can be divisors of a specific number limited quantity, but there are an infinite number of multiples.

Common multiple natural numbers- a number that is divisible by them without a remainder.

How to find the least common multiple of numbers

The least common multiple (LCM) of numbers (two, three or more) is the smallest natural number that is divisible by all of these numbers.

To find the LOC, you can use several methods.

For small numbers, it is convenient to write down all the multiples of these numbers on a line until you find something common among them. Multiples are denoted by the capital letter K.

For example, multiples of 4 can be written like this:

K (4) = (8,12, 16, 20, 24, ...)

K (6) = (12, 18, 24, ...)

Thus, you can see that the least common multiple of the numbers 4 and 6 is the number 24. This notation is done as follows:

LCM(4, 6) = 24

If the numbers are large, find the common multiple of three or more numbers, then it is better to use another method of calculating the LCM.

To complete the task, you need to factor the given numbers into prime factors.

First you need to write down the decomposition of the largest number on a line, and below it - the rest.

In the expansion of each number there may be different quantity multipliers.

For example, let's factor the numbers 50 and 20 into prime factors.

In the expansion of the smaller number, it is necessary to emphasize the factors that are absent in the expansion of the first one. large number, and then add them to it. In the example presented, a two is missing.

Now you can calculate the least common multiple of 20 and 50.

LCM(20, 50) = 2 * 5 * 5 * 2 = 100

So, the product of prime factors more and the factors of the second number that were not included in the expansion of the larger number will be the least common multiple.

To find the LCM of three or more numbers, you should factor them all into prime factors, as in the previous case.

As an example, you can find the least common multiple of the numbers 16, 24, 36.

36 = 2 * 2 * 3 * 3

24 = 2 * 2 * 2 * 3

16 = 2 * 2 * 2 * 2

Thus, only two twos from the expansion of sixteen were not included in the factorization of a larger number (one is in the expansion of twenty-four).

Thus, they need to be added to the expansion of a larger number.

LCM(12, 16, 36) = 2 * 2 * 3 * 3 * 2 * 2 = 9

There are special cases of determining the least common multiple. So, if one of the numbers can be divided without a remainder by another, then the larger of these numbers will be the least common multiple.

For example, the LCM of twelve and twenty-four is twenty-four.

If you need to find the least common multiple of each other prime numbers, which do not have identical divisors, then their LCM will be equal to their product.

For example, LCM (10, 11) = 110.