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Knock three numbers rule. How to find the least common multiple of numbers. Least common multiple of three or more numbers

But many integers are evenly divisible by other natural numbers.

For example:

The number 12 is divisible by 1, by 2, by 3, by 4, by 6, by 12;

The number 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 divisible (for 12 it is 1, 2, 3, 4, 6 and 12) are called number divisors. Divisor of a natural number a is the natural number that divides the given number a without a trace. A natural number that has more than two factors is called composite .

Note that the numbers 12 and 36 have common divisors. These are the numbers: 1, 2, 3, 4, 6, 12. The largest divisor of these numbers is 12. The common divisor of these two numbers a and b is the number by which both given numbers are divisible without a remainder a and b.

common multiple several numbers is called the number that is divisible by each of these numbers. For example, the numbers 9, 18 and 45 have a common multiple of 180. But 90 and 360 are also their common multiples. Among all jcommon multiples, there is always the smallest one, in this case it is 90. This number is called leastcommon multiple (LCM).

LCM is always a natural number, which must be greater than the largest of the numbers for which it is defined.

Least common multiple (LCM). Properties.

Commutativity:

Associativity:

In particular, if and are coprime numbers , then:

Least common multiple of two integers m and n is a divisor of all other common multiples m and n. Moreover, the set of common multiples m,n coincides with the set of multiples for LCM( m,n).

The asymptotics for can be expressed in terms of some number-theoretic functions.

So, Chebyshev function. As well as:

This follows from the definition and properties of the Landau function g(n).

What follows from the law of distribution of prime numbers.

Finding the least common multiple (LCM).

NOC( a, b) can be calculated in several ways:

1. If the greatest common divisor is known, you can use its relationship with the LCM:

2. Let the canonical expansion of both numbers into prime factors:

where p 1 ,...,p k are various prime numbers, and d 1 ,...,d k and e 1 ,...,ek are non-negative integers (they can be zero if the corresponding prime is not in the decomposition).

Then LCM ( a,b) is calculated by the formula:

In other words, the LCM expansion contains all prime factors that are included in at least one of the number expansions a, b, and the largest of the two exponents of this factor is taken.

Example:

The calculation of the least common multiple of several numbers can be reduced to several successive calculations of the LCM of two numbers:

Rule. To find the LCM of a series of numbers, you need:

- decompose numbers into prime factors;

- transfer the largest expansion into the factors of the desired product (the product of the factors of the a large number from the given ones), and then add factors from the decomposition of other numbers that do not occur in the first number or are in it a smaller number of times;

- the resulting product of prime factors will be the LCM of the given numbers.

Any two or more natural numbers have their own LCM. If the numbers are not multiples of each other or do not have the same factors in the expansion, then their LCM is equal to the product of these numbers.

The prime factors of the number 28 (2, 2, 7) were supplemented with a factor of 3 (the number 21), the resulting product (84) will be the smallest number that is divisible by 21 and 28.

The prime factors of the largest number 30 were supplemented with a factor of 5 of the number 25, the resulting product 150 is greater than the largest number 30 and is divisible by all given numbers without a remainder. This is the smallest possible product (150, 250, 300...) that all given numbers are multiples of.

The numbers 2,3,11,37 are prime, so their LCM is equal to the product of the given numbers.

rule. To calculate the LCM of prime numbers, you need to multiply all these numbers together.

Another option:

To find the least common multiple (LCM) of several numbers you need:

1) represent each number as a product of its prime factors, for example:

504 \u003d 2 2 2 3 3 7,

2) write down the powers of all prime factors:

504 \u003d 2 2 2 3 3 7 \u003d 2 3 3 2 7 1,

3) write down all prime divisors (multipliers) of each of these numbers;

4) choose the largest degree of each of them, found in all expansions of these numbers;

5) multiply these powers.

Example. Find the LCM of numbers: 168, 180 and 3024.

Solution. 168 \u003d 2 2 2 3 7 \u003d 2 3 3 1 7 1,

180 \u003d 2 2 3 3 5 \u003d 2 2 3 2 5 1,

3024 = 2 2 2 2 3 3 3 7 = 2 4 3 3 7 1 .

We write out the largest powers of all prime divisors and multiply them:

LCM = 2 4 3 3 5 1 7 1 = 15120.

Numbers that are divisible by 10 are called multiples of 10. For example, 30 or 50 are multiples of 10. 28 is a multiple of 14. Numbers that are divisible by both 10 and 14 are naturally called common multiples of 10 and 14.

We can find any number of common multiples. For example, 140, 280, etc.

The natural question is: how to find the least common multiple, the least common multiple?

Of the multiples found for 10 and 14, the smallest so far is 140. But is it the least common multiple?

Let's factor our numbers:

Let's construct a number that is divisible by 10 and 14. To be divisible by 10, you need to have factors 2 and 5. To be divisible by 14, you need to have factors 2 and 7. But 2 is already there, it remains to add 7. The resulting number 70 is the common multiple of 10 and 14. In this case, it will not be possible to construct a number less than this so that it is also a common multiple.

So this is what it is least common multiple. We use the notation LCM for it.

Let's find GCD and LCM for numbers 182 and 70.

Calculate yourself:

We check:

To understand what GCD and LCM are, one cannot do without factoring. But, when we already understood what it is, it is no longer necessary to factor it every time.

For example:

You can easily see that for two numbers where one is divisible by the other, the smaller one is their GCD and the larger one is their LCM. Try to explain why this is so.

Dad's step length is 70 cm, and the little daughter's step is 15 cm. They start walking with their feet on the same mark. How far will they walk before their feet are level again?

Dad and daughter start moving. First, the legs are on the same mark. After walking a few steps, their legs again stood on the same mark. This means that both dad and daughter got a whole number of steps to this mark. This means that the distance to her should be divided by the step length of both dad and daughter.

That is, we must find:

That is, it will happen in 210 cm = 2 m 10 cm.

It is easy to understand that the father will take 3 steps, and the daughter - 14 (Fig. 1).

Rice. 1. Illustration for the problem

Task 1

Petya has 100 friends on VKontakte, and Vanya has 200. How many friends does Petya and Vanya have together if there are 30 friends in common?

Answer 300 is incorrect, because they may have mutual friends.

Let's solve this problem like this. Let's depict the set of all Petya's friends around. Let's depict many of Vanya's friends in a different circle, more.

These circles have a common part. There are common friends there. This a common part called the "intersection" of the two sets. That is, the set of mutual friends is the intersection of the sets of friends of each.

Rice. 2. Circles of many friends

If there are 30 common friends, then on the left 70 are only Petina's friends, and 170 are Vanina's only (see Fig. 2).

How much?

An entire large set consisting of two circles is called the union of the two sets.

In fact, VK itself solves the problem of crossing two sets for us, it immediately indicates a lot of mutual friends when you go to the page of another person.

The situation with GCD and LCM of two numbers is very similar.

Task 2

Consider two numbers: 126 and 132.

We will depict their prime factors in circles (see Fig. 3).

Rice. 3. Circles with prime factors

The intersection of sets are common divisors. Of these, NOD consists.

The union of the two sets gives us the LCM.

Bibliography

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.

2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium. 2006.

3. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - M.: Enlightenment, 1989.

4. Rurukin A.N., Chaikovsky I.V. Tasks for the course of mathematics grade 5-6. - M.: ZSh MEPhI, 2011.

5. Rurukin A.N., Sochilov S.V., Chaikovsky K.G. Mathematics 5-6. A manual for students of the 6th grade of the MEPhI correspondence school. - M.: ZSh MEPhI, 2011.

6. Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Interlocutor textbook for grades 5-6 high school. - M .: Education, Mathematics Teacher Library, 1989.

3. Website "School Assistant" ()

Homework

1. Three tourist boat trips start in the port city, the first of which lasts 15 days, the second - 20 and the third - 12 days. Returning to the port, the ships on the same day again go on a voyage. Motor ships left the port on all three routes today. In how many days will they sail together for the first time? How many trips will each ship make?

2. Find the LCM of numbers:

3. Find the prime factors of the least common multiple of numbers:

And if: , , .

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.

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 all prime factors that are simultaneously present in the factorizations of the given 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 3 and 5 common to both numbers, we get the product the following kind: 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 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;
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 .

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, 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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The topic "Multiple numbers" is studied in grade 5 secondary school. Its goal is to improve written and oral skills 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.

The least common multiple of two numbers is directly related to the greatest common divisor of those numbers. This link between GCD and 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 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.

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.

Denote gcd(a, b) as d . Then we can write down 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 properties of divisibility, is equivalent to the condition that a 1 k is divisible by b one .

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

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

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

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

The rationale for this fact is quite 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 successively 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 numbers m k-1 and a k , therefore, coincide with multiples of m k . And since the least positive multiple of the number m k is the number m k itself, then the least common multiple of the numbers a 1 , a 2 , …, a k is m k .

Bibliography.

Vilenkin N.Ya. etc. 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: Tutorial for students of physics and mathematics. specialties of pedagogical institutes.