How to find the common multiple of numbers. Divisors and multiples

How to find LCM (least common multiple)

The common multiple of two integers is the integer that is evenly divisible by both given numbers without remainder.

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 delete from the resulting rows prime factors the same numbers. 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 multiplier 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.

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 when 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 required 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 you should use a different technique here, which 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 expansion of the number 16 were not included in the decomposition 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 with further study 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.

Consider three ways to find the least common multiple.

Finding by Factoring

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

Suppose we need to find the LCM of numbers: 99, 30 and 28. To do this, we decompose 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 highest occurring power and multiply them together:

2 2 3 2 5 7 11 = 13 860

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

To find the least common multiple of given numbers, you need to decompose them into prime factors, then take each prime factor with the largest exponent with which it occurs, and multiply these factors together.

Since coprime numbers have no common prime factors, their least common multiple is equal to the product of these numbers. For example, three numbers: 20, 49 and 33 are coprime. That's why

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

The same should be done when looking for the least common multiple of various prime numbers. For example, LCM (3, 7, 11) = 3 7 11 = 231.

Finding by selection

The second way is to find the least common multiple by fitting.

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

NOC(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 numbers that are multiples of the largest number, multiplying it by natural numbers in ascending order and checking whether the remaining given numbers are divisible by the resulting product.

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

24 1 = 24 is divisible by 3 but not divisible by 18.

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

24 3 \u003d 72 - divisible by 3 and 18.

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

Finding by Sequential Finding LCM

The third way is to find the least common multiple by successively 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 into their GCD:

So LCM(12, 8) = 24.

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

  1. First, the LCM of any two of the given numbers is found.
  2. Then, the 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, and so on.
  4. Thus the LCM search continues as long as there are numbers.

Example 2. Let's find the LCM of three given numbers: 12, 8 and 9. We have 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 24 and the third given number - 9. Determine their greatest common divisor: gcd (24, 9) = 3. Multiply LCM with the number 9:

We divide the product into their GCD:

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

But many natural numbers 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 decomposition of both numbers into prime factors be known:

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 expansion).

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 NOC. 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 trace. 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.

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