The topic "Multiple numbers" 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 - "multiple numbers" and "divisors", the technique of finding divisors and multiples of a natural number, the ability to find LCM in various ways is 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.

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

It is necessary 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.

When calculating the LCM, there are special cases.

1. If you need to find a common multiple for 2 numbers (for example, 80 and 20), where one of them (80) is divisible 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.

Consider the last example. 6 and 7 in relation to 42 are divisors. They divide a multiple without a remainder.

In this example, 6 and 7 are pair divisors. 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.

In another example, you need to determine if 9 is a divisor with respect to 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 is itself divisible by that 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 as a product of powers:

168=2³x3¹x7¹

2⁴х3³х5¹х7¹=15120

LCM (168, 180, 3024) = 15120.

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 look at 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.

Decision

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.

Decision

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 will become 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.

Decision

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.

Decision

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 consistently 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 .

Decision

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

Decision

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 .

Decision

Let's change the numbers − 145 and − 45 to their opposite numbers 145 and 45 . Now, using 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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Students are given a lot of math assignments. Among them, very often there are tasks with the following formulation: there are two values. 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 with different denominators. In the 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 define the term multiple. Most often, the wording of this concept is as follows: a multiple of some value A is a natural number that will be divisible by A without a remainder. So, for 4, 8, 12, 16, 20 and so on, up to the necessary 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 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 fully divisible by all the given numbers.

There are several ways to find such a value. Let's consider the following methods:

1. If the numbers are small, then write in the line all divisible by it. 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 these are large or you need to find a multiple for 3 or more values, then here you should use a different technique that involves decomposing numbers into prime factors. First, lay out the largest of the indicated, then all the rest. 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 of them, underline the factors and add to the largest. The result will be 100, which will be the least 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's expand each of them: 16 = 2*2*2*2, 24=2*2*2*3, 36=2*2*3*3. Only two deuces from the decomposition 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 is the general technique for finding the smallest value for two, three or more values. However, there are also private methods, helping to search for NOCs, if the previous ones do not help.

How to find GCD and NOC.

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 divisible by the others without a remainder, then the lowest multiple of these numbers is equal to it (NOC 60 and 15 is equal to 15);
• Coprime numbers do not have 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 cases of decomposition of composite numbers, which are the subject of separate articles and even Ph.D. dissertations.

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

Some examples

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

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

Finding the NOC is much easier than it might seem at first. For this, both a simple expansion and the multiplication of simple values ​​\u200b\u200bto each other are used.. The ability to work with this section of mathematics helps in the further study of mathematical topics, especially fractions of varying degrees of complexity.

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

Video

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

How to find LCM (least common multiple)

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

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 the numbers 6 and 9 will be 18.

This method is convenient when both numbers are small and it is easy to multiply them by a sequence of integers. However, there are cases when you need to find the LCM for two-digit or three-digit numbers, and also when there are three or even more initial numbers.

Method 2. You can find the LCM by decomposing the original numbers into prime factors.
After decomposition, it is necessary to cross out the same numbers from the resulting series of prime factors. The remaining numbers of the first number will be the factor for the second, and the remaining numbers of the second number will be the factor for the first.

Example for the number 75 and 60.
The least common multiple of the numbers 75 and 60 can be found without writing out multiples of these numbers in a row. To do this, we decompose 75 and 60 into prime factors:
75 = 3 * 5 * 5, and
60 = 2 * 2 * 3 * 5 .
As you can see, the factors 3 and 5 occur in both rows. Mentally we "cross out" them.
Let's write down the remaining factors included in the expansion of each of these numbers. When decomposing the number 75, we left the number 5, and when decomposing the number 60, we left 2 * 2
So, 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 the numbers remaining from the expansion of the number 60 (this is 2 * 2) multiply by 75. That is, for ease of understanding , we say that we multiply "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 LCM for numbers 12, 16, 24
In this case, our actions will be somewhat more complicated. But, first, as always, we decompose all numbers into prime factors
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 successively go through its factors, crossing them out if at least one of the other rows of numbers has the same factor that has not yet been crossed out.

Step 1 . We see that 2 * 2 occurs in all series of numbers. We 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 action is expected for the number 16.
12 = 2 * 2 * 3
16 = 2 * 2 * 2 * 2
24 = 2 * 2 * 2 * 3

As you can see, when decomposing the number 12, we "crossed out" all the numbers. So the finding of the NOC is completed. It remains only to calculate its value.
For the number 12, we take the remaining factors from the number 16 (the closest 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 ways of finding the LCM are correct.

Greatest Common Divisor

Definition 2

If a natural number a is divisible by a natural number $b$, then $b$ is called a divisor of $a$, and the number $a$ is called a multiple of $b$.

Let $a$ and $b$ be natural numbers. The number $c$ is called a common divisor for both $a$ and $b$.

The set of common divisors of the numbers $a$ and $b$ is finite, since none of these divisors can be greater than $a$. This means that among these divisors there is the largest one, which is called the greatest common divisor of the numbers $a$ and $b$, and the notation is used to denote it:

$gcd \ (a;b) \ ​​or \ D \ (a;b)$

To find the greatest common divisor of two numbers:

1. Find the product of the numbers found in step 2. The resulting number will be the desired greatest common divisor.

Example 1

Find the gcd of the numbers $121$ and $132.$

$242=2\cdot 11\cdot 11$

$132=2\cdot 2\cdot 3\cdot 11$

Choose the numbers that are included in the expansion of these numbers

$242=2\cdot 11\cdot 11$

$132=2\cdot 2\cdot 3\cdot 11$

Find the product of the numbers found in step 2. The resulting number will be the desired greatest common divisor.

$gcd=2\cdot 11=22$

Example 2

Find the GCD of monomials $63$ and $81$.

We will find according to the presented algorithm. For this:

Let's decompose numbers into prime factors

$63=3\cdot 3\cdot 7$

$81=3\cdot 3\cdot 3\cdot 3$

We select the numbers that are included in the expansion of these numbers

$63=3\cdot 3\cdot 7$

$81=3\cdot 3\cdot 3\cdot 3$

Let's find the product of the numbers found in step 2. The resulting number will be the desired greatest common divisor.

$gcd=3\cdot 3=9$

You can find the GCD of two numbers in another way, using the set of divisors of numbers.

Example 3

Find the gcd of the numbers $48$ and $60$.

Decision:

Find the set of divisors of $48$: $\left\((\rm 1,2,3.4.6,8,12,16,24,48)\right\)$

Now let's find the set of divisors of $60$:$\ \left\((\rm 1,2,3,4,5,6,10,12,15,20,30,60)\right\)$

Let's find the intersection of these sets: $\left\((\rm 1,2,3,4,6,12)\right\)$ - this set will determine the set of common divisors of the numbers $48$ and $60$. The largest element in this set will be the number $12$. So the greatest common divisor of $48$ and $60$ is $12$.

Definition of NOC

Definition 3

common multiple of natural numbers$a$ and $b$ is a natural number that is a multiple of both $a$ and $b$.

Common multiples of numbers are numbers that are divisible by the original without a remainder. For example, for the numbers $25$ and $50$, the common multiples will be the numbers $50,100,150,200$, etc.

The least common multiple will be called the least common multiple and denoted by LCM$(a;b)$ or K$(a;b).$

To find the LCM of two numbers, you need:

1. Decompose numbers into prime factors
2. Write out the factors that are part of the first number and add to them the factors that are part of the second and do not go to the first

Example 4

Find the LCM of the numbers $99$ and $77$.

We will find according to the presented algorithm. For this

Decompose numbers into prime factors

$99=3\cdot 3\cdot 11$

Write down the factors included in the first

add to them factors that are part of the second and do not go to the first

Find the product of the numbers found in step 2. The resulting number will be the desired least common multiple

$LCC=3\cdot 3\cdot 11\cdot 7=693$

Compiling lists of divisors of numbers is often very time consuming. There is a way to find GCD called Euclid's algorithm.

Statements on which Euclid's algorithm is based:

If $a$ and $b$ are natural numbers, and $a\vdots b$, then $D(a;b)=b$

If $a$ and $b$ are natural numbers such that $b

Using $D(a;b)= D(a-b;b)$, we can successively decrease the numbers under consideration until we reach a pair of numbers such that one of them is divisible by the other. Then the smaller of these numbers will be the desired greatest common divisor for the numbers $a$ and $b$.

Properties of GCD and LCM

1. Any common multiple of $a$ and $b$ is divisible by K$(a;b)$
2. If $a\vdots b$ , then K$(a;b)=a$

If K$(a;b)=k$ and $m$-natural number, then K$(am;bm)=km$

If $d$ is a common divisor for $a$ and $b$, then K($\frac(a)(d);\frac(b)(d)$)=$\ \frac(k)(d) $

If $a\vdots c$ and $b\vdots c$ , then $\frac(ab)(c)$ is a common multiple of $a$ and $b$

For any natural numbers $a$ and $b$ the equality

$D(a;b)\cdot K(a;b)=ab$

Any common divisor of $a$ and $b$ is a divisor of $D(a;b)$