How to find a common multiple. Least Common Multiple (LCM) – Definition, Examples and Properties. General scheme for finding the least common multiple

Let's find the greatest common divisor of GCD (36; 24)

Solution steps

Method No. 1

36 - composite number
24 - composite number

Let's expand the number 36

36: 2 = 18
18: 2 = 9 - divisible by the prime number 2
9: 3 = 3 - divisible by the prime number 3.

Let's break down the number 24 into prime factors and highlight them in green. We begin to select a divisor from prime numbers, starting with the smallest prime number 2, until the quotient turns out to be a prime number

24: 2 = 12 - divisible by the prime number 2
12: 2 = 6 - divisible by the prime number 2
6: 2 = 3
We complete the division since 3 is a prime number

2) Highlight it in blue and write out the common factors

36 = 2 ⋅ 2 ⋅ 3 ⋅ 3
24 = 2 ⋅ 2 ⋅ 2 ⋅ 3
Common factors (36; 24): 2, 2, 3

3) Now, to find the GCD you need to multiply the common factors

Answer: GCD (36; 24) = 2 ∙ 2 ∙ 3 ​​= 12

Method No. 2

1) Find all possible divisors of the numbers (36; 24). To do this, we will alternately divide the number 36 into divisors from 1 to 36, and the number 24 into divisors from 1 to 24. If the number is divisible without a remainder, then we write the divisor in the list of divisors.

For number 36
36: 1 = 36; 36: 2 = 18; 36: 3 = 12; 36: 4 = 9; 36: 6 = 6; 36: 9 = 4; 36: 12 = 3; 36: 18 = 2; 36: 36 = 1;

For the number 24 Let's write down all the cases when it is divisible without a remainder:
24: 1 = 24; 24: 2 = 12; 24: 3 = 8; 24: 4 = 6; 24: 6 = 4; 24: 8 = 3; 24: 12 = 2; 24: 24 = 1;

2) Let’s write down all the common divisors of the numbers (36; 24) and highlight the largest one in green, this will be the greatest common divisor of the gcd of the numbers (36; 24)

Common factors of numbers (36; 24): 1, 2, 3, 4, 6, 12

Answer: GCD (36 ; 24) = 12

Let's find the least common multiple of the LCM (52; 49)

Solution steps

Method No. 1

1) Let's factor the numbers into prime factors. To do this, let’s check whether each of the numbers is prime (if a number is prime, then it cannot be decomposed into prime factors, and it is itself a decomposition)

52 - composite number
49 - composite number

Let's expand the number 52 into prime factors and highlight them in green. We begin to select a divisor from prime numbers, starting with the smallest prime number 2, until the quotient turns out to be a prime number

52: 2 = 26 - divisible by the prime number 2
26: 2 = 13 - divisible by the prime number 2.
We complete the division since 13 is a prime number

Let's expand the number 49 into prime factors and highlight them in green. We begin to select a divisor from prime numbers, starting with the smallest prime number 2, until the quotient turns out to be a prime number

49: 7 = 7 - divisible by the prime number 7.
We complete the division since 7 is a prime number

2) First of all, write down the factors of the largest number, and then the smaller number. Let's find the missing factors, highlight in blue in the expansion of the smaller number the factors that were not included in the expansion of the larger number.

52 = 2 ∙ 2 ∙ 13
49 = 7 ∙ 7

3) Now, to find the LCM you need to multiply the factors of the larger number with the missing factors, which are highlighted in blue

LCM (52 ; 49) = 2 ∙ 2 ∙ 13 ∙ 7 ∙ 7 = 2548

Method No. 2

1) Find all possible multiples of the numbers (52; 49). To do this, we will alternately multiply the number 52 by the numbers from 1 to 49, and the number 49 by the numbers from 1 to 52.

Select all multiples 52 in green:

52 ∙ 1 = 52 ; 52 ∙ 2 = 104 ; 52 ∙ 3 = 156 ; 52 ∙ 4 = 208 ;
52 ∙ 5 = 260 ; 52 ∙ 6 = 312 ; 52 ∙ 7 = 364 ; 52 ∙ 8 = 416 ;
52 ∙ 9 = 468 ; 52 ∙ 10 = 520 ; 52 ∙ 11 = 572 ; 52 ∙ 12 = 624 ;
52 ∙ 13 = 676 ; 52 ∙ 14 = 728 ; 52 ∙ 15 = 780 ; 52 ∙ 16 = 832 ;
52 ∙ 17 = 884 ; 52 ∙ 18 = 936 ; 52 ∙ 19 = 988 ; 52 ∙ 20 = 1040 ;
52 ∙ 21 = 1092 ; 52 ∙ 22 = 1144 ; 52 ∙ 23 = 1196 ; 52 ∙ 24 = 1248 ;
52 ∙ 25 = 1300 ; 52 ∙ 26 = 1352 ; 52 ∙ 27 = 1404 ; 52 ∙ 28 = 1456 ;
52 ∙ 29 = 1508 ; 52 ∙ 30 = 1560 ; 52 ∙ 31 = 1612 ; 52 ∙ 32 = 1664 ;
52 ∙ 33 = 1716 ; 52 ∙ 34 = 1768 ; 52 ∙ 35 = 1820 ; 52 ∙ 36 = 1872 ;
52 ∙ 37 = 1924 ; 52 ∙ 38 = 1976 ; 52 ∙ 39 = 2028 ; 52 ∙ 40 = 2080 ;
52 ∙ 41 = 2132 ; 52 ∙ 42 = 2184 ; 52 ∙ 43 = 2236 ; 52 ∙ 44 = 2288 ;
52 ∙ 45 = 2340 ; 52 ∙ 46 = 2392 ; 52 ∙ 47 = 2444 ; 52 ∙ 48 = 2496 ;
52 ∙ 49 = 2548 ;

Select all multiples 49 in green:

49 ∙ 1 = 49 ; 49 ∙ 2 = 98 ; 49 ∙ 3 = 147 ; 49 ∙ 4 = 196 ;
49 ∙ 5 = 245 ; 49 ∙ 6 = 294 ; 49 ∙ 7 = 343 ; 49 ∙ 8 = 392 ;
49 ∙ 9 = 441 ; 49 ∙ 10 = 490 ; 49 ∙ 11 = 539 ; 49 ∙ 12 = 588 ;
49 ∙ 13 = 637 ; 49 ∙ 14 = 686 ; 49 ∙ 15 = 735 ; 49 ∙ 16 = 784 ;
49 ∙ 17 = 833 ; 49 ∙ 18 = 882 ; 49 ∙ 19 = 931 ; 49 ∙ 20 = 980 ;
49 ∙ 21 = 1029 ; 49 ∙ 22 = 1078 ; 49 ∙ 23 = 1127 ; 49 ∙ 24 = 1176 ;
49 ∙ 25 = 1225 ; 49 ∙ 26 = 1274 ; 49 ∙ 27 = 1323 ; 49 ∙ 28 = 1372 ;
49 ∙ 29 = 1421 ; 49 ∙ 30 = 1470 ; 49 ∙ 31 = 1519 ; 49 ∙ 32 = 1568 ;
49 ∙ 33 = 1617 ; 49 ∙ 34 = 1666 ; 49 ∙ 35 = 1715 ; 49 ∙ 36 = 1764 ;
49 ∙ 37 = 1813 ; 49 ∙ 38 = 1862 ; 49 ∙ 39 = 1911 ; 49 ∙ 40 = 1960 ;
49 ∙ 41 = 2009 ; 49 ∙ 42 = 2058 ; 49 ∙ 43 = 2107 ; 49 ∙ 44 = 2156 ;
49 ∙ 45 = 2205 ; 49 ∙ 46 = 2254 ; 49 ∙ 47 = 2303 ; 49 ∙ 48 = 2352 ;
49 ∙ 49 = 2401 ; 49 ∙ 50 = 2450 ; 49 ∙ 51 = 2499 ; 49 ∙ 52 = 2548 ;

2) Let’s write down all the common multiples of the numbers (52; 49) and highlight the smallest one in green, this will be the smallest common multiple of the numbers (52; 49).

Common multiples of numbers (52; 49): 2548

Answer: LCM (52; 49) = 2548

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 with different denominators. In this article we will look at how to find LOC and basic concepts.

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 like this: 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.

In this case, 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:

1. 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.
2. 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.
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 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 the general technique is for 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:

1. 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.
2. 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.
3. 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 of simple values ​​by each other are used. The ability to work with this section of mathematics helps with further study of mathematical topics, especially fractions of varying degrees of complexity.

Don’t forget to periodically solve examples using different methods; this develops your 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.

Common multiples

Simply put, any integer that is divisible by each of the given numbers is common multiple given integers.

You can find the common multiple of two or more integers.

Example 1

Calculate the common multiple of two numbers: $2$ and $5$.

Solution.

By definition, the common multiple of $2$ and $5$ is $10$, because it is a multiple of the number $2$ and the number $5$:

Common multiples of the numbers $2$ and $5$ will also be the numbers $–10, 20, –20, 30, –30$, etc., because all of them are divided into numbers $2$ and $5$.

Note 1

Zero is a common multiple of any number of non-zero integers.

According to the properties of divisibility, if a certain number is a common multiple of several numbers, then the number opposite in sign will also be a common multiple of the given numbers. This can be seen from the example considered.

For given integers, you can always find their common multiple.

Example 2

Calculate the common multiple of $111$ and $55$.

Solution.

Let's multiply the given numbers: $111\div 55=6105$. It is easy to verify that the number $6105$ is divisible by the number $111$ and the number $55$:

$6105\div 111=$55;

$6105\div 55=$111.

Thus, $6105$ is a common multiple of $111$ and $55$.

Answer: The common multiple of $111$ and $55$ is $6105$.

But, as we have already seen from the previous example, this common multiple is not one. Other common multiples would be $–6105, 12210, –12210, 61050, –61050$, etc. Thus, we came to the following conclusion:

Note 2

Any set of integers has an infinite number of common multiples.

In practice, they are limited to finding common multiples of only positive integer (natural) numbers, because the sets of multiples of a given number and its opposite coincide.

Determining Least Common Multiple

Of all multiples of given numbers, the least common multiple (LCM) is used most often.

Definition 2

The least positive common multiple of given integers is least common multiple these numbers.

Example 3

Calculate the LCM of the numbers $4$ and $7$.

Solution.

Because these numbers have no common divisors, then $LCM(4,7)=28$.

Answer: $NOK (4,7)=28$.

Finding NOC via GCD

Because there is a connection between LCM and GCD, with its help you can calculate LCM of two positive integers:

Note 3

Example 4

Calculate the LCM of the numbers $232$ and $84$.

Solution.

Let's use the formula to find the LCM through GCD:

$LCD (a,b)=\frac(a\cdot b)(GCD (a,b))$

Let's find the GCD of the numbers $232$ and $84$ using the Euclidean algorithm:

$232=84\cdot 2+64$,

$84=64\cdot 1+20$,

$64=20\cdot 3+4$,

Those. $GCD(232, 84)=4$.

Let's find $LCC (232, 84)$:

$NOK (232.84)=\frac(232\cdot 84)(4)=58\cdot 84=4872$

Answer: $NOK (232.84)=$4872.

Example 5

Compute $LCD(23, 46)$.

Solution.

Because $46$ is divisible by $23$, then $gcd (23, 46)=23$. Let's find the LOC:

$NOK (23.46)=\frac(23\cdot 46)(23)=46$

Answer: $NOK (23.46)=$46.

Thus, one can formulate rule:

Note 4

We call numbers that are divisible by 10 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 as many common multiples as we want. For example, 140, 280, etc.

A natural question is: how to find the smallest 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 of 2 and 5. To be divisible by 14, you need to have factors of 2 and 7. But 2 is already there, all you have to do is add 7. The resulting number 70 is the common multiple of 10 and 14. However, it will not be possible to construct a number smaller than this so that it is also a common multiple.

So this is it least common multiple. For this we use the notation NOC.

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

Calculate for yourself:

3.

We check:

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

For example:

You can easily verify 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 yourself why this is so.

The step length of a dad is 70 cm, and that of a little daughter is 15 cm. They start walking with their feet on the same mark. How far will they walk before their legs are level again?

Dad and daughter start moving. At first, the legs are on the same mark. After walking a few steps, their feet returned to the same level. This means that both dad and daughter got a whole number of steps to reach this mark. This means that the distance to her should be divided by the step length of both father and daughter.

That is, we must find:

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

It is not difficult to understand that the father will take 3 steps, and the daughter will take 14 (Fig. 1).

Rice. 1. Illustration for the problem

Problem 1

Petya has 100 friends on the VKontakte network, and Vanya has 200. How many friends do Petya and Vanya have together, if they have 30 mutual friends?

Answer 300 is incorrect because they may have mutual friends.

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

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

Rice. 2. Circles of many friends

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

How much in total?

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

In fact, VK itself solves the problem of intersection of two sets for us; it immediately indicates many mutual friends when you visit another person’s page.

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

Problem 2

Consider two numbers: 126 and 132.

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

Rice. 3. Circles with prime factors

The intersection of sets is their common divisors. GCD consists of them.

The union of 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.: Education, 1989.

4. Rurukin A.N., Tchaikovsky I.V. Assignments for the mathematics course for grades 5-6. - M.: ZSh MEPhI, 2011.

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

6. Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for 5-6 grades of secondary school. - M.: Education, Mathematics Teacher Library, 1989.

3. Website “School Assistant” ()

Homework

1. Three tourist boat voyages begin in the port city, the first of which lasts 15 days, the second - 20 and the third - 12 days. Having returned to the port, the ships set off again on the same day. Today, ships left the port on all three routes. In how many days will they go sailing together again for the first time? How many trips will each ship make?

2. Find the LCM of the numbers:

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

And if: , , .

Let's consider solving the following problem. The boy's step is 75 cm, and the girl's step is 60 cm. It is necessary to find the smallest distance at which they both take an integer number of steps.

Solution. The entire path that the guys will go through must be divisible by 60 and 70, since they must each take an integer number of steps. In other words, the answer must be a multiple of both 75 and 60.

First, we will write down all the multiples of the number 75. We get:

• 75, 150, 225, 300, 375, 450, 525, 600, 675, … .

Now let's write down the numbers that will be multiples of 60. We get:

• 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, 660, … .

Now we find the numbers that are in both rows.

• Common multiples of numbers would be 300, 600, etc.

The smallest of them is the number 300. In this case, it will be called the least common multiple of the numbers 75 and 60.

Returning to the condition of the problem, the smallest distance at which the guys will take an integer number of steps will be 300 cm. The boy will cover this path in 4 steps, and the girl will need to take 5 steps.

Determining Least Common Multiple

• The least common multiple of two natural numbers a and b is the smallest natural number that is a multiple of both a and b.

In order to find the least common multiple of two numbers, it is not necessary to write down all the multiples of these numbers in a row.

You can use the following method.

How to find the least common multiple

First you need to factor these numbers into prime factors.

• 60 = 2*2*3*5,
• 75=3*5*5.

Now let’s write down all the factors that are in the expansion of the first number (2,2,3,5) and add to it all the missing factors from the expansion of the second number (5).

As a result, we get a series of prime numbers: 2,2,3,5,5. The product of these numbers will be the least common factor for these numbers. 2*2*3*5*5 = 300.

General scheme for finding the least common multiple

• 1. Divide numbers into prime factors.
• 2. Write down the prime factors that are part of one of them.
• 3. Add to these factors all those that are in the expansion of the others, but not in the selected one.
• 4. Find the product of all the written factors.

This method is universal. It can be used to find the least common multiple of any number of natural numbers.