How to determine the least common multiple of numbers. Methods for finding the least common multiple, nok - this, and all the explanations

How to find LCM (least common multiple)

A common multiple of two integers is an integer that is evenly divisible by both given numbers without leaving a remainder.

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

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 numbers 6 and 9 will be equal to 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 factoring the original numbers into prime factors.
After decomposition, it is necessary to cross out identical numbers from the resulting series of prime factors. The remaining numbers of the first number will be a multiplier for the second, and the remaining numbers of the second will be a multiplier for the first.

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

Step 1. We see that 2 * 2 occurs in all series of numbers. Let's 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 actions are 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. This means that the finding of the LOC is completed. All that remains is to calculate its value.
For the number 12, take the remaining factors of the number 16 (next 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 methods of finding the LCM are correct.

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.

1. 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.

2. 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.

3. Find the smallest number that is present in both sets of multiples. You may have to write long series of multiples to find the total number. The smallest number that is present in both sets of multiples is the least common multiple.

   • For example, the smallest number that appears 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 result in 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 the factorization 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 factor is 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 the next two 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.

The material presented below is a logical continuation of the theory from the article entitled LCM - least common multiple, definition, examples, connection between LCM and GCD. Here we will talk about finding the least common multiple (LCM), and we will pay special attention to solving examples. First, we will show how the LCM of two numbers is calculated using the GCD of these numbers. Next, we'll look at finding the least common multiple by factoring numbers into prime factors. After this, we will focus on finding the LCM of three or more numbers, and also pay attention to calculating the LCM of negative numbers.

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Calculating Least Common Multiple (LCM) via GCD

One way to find the least common multiple is based on the relationship between LCM and GCD. The existing connection between LCM and GCD allows us to calculate the least common multiple of two positive integers through a known greatest common divisor. The corresponding formula is LCM(a, b)=a b:GCD(a, b) . Let's look at examples of finding the LCM using the given formula.

Example.

Find the least common multiple of two numbers 126 and 70.

Solution.

In this example a=126 , b=70 . Let us use the connection between LCM and GCD, expressed by the formula LCM(a, b)=a b:GCD(a, b). That is, first we have to find the greatest common divisor of the numbers 70 and 126, after which we can calculate the LCM of these numbers using the written formula.

Let's find GCD(126, 70) using the Euclidean algorithm: 126=70·1+56, 70=56·1+14, 56=14·4, therefore, GCD(126, 70)=14.

Now we find the required least common multiple: GCD(126, 70)=126·70:GCD(126, 70)= 126·70:14=630.

Answer:

LCM(126, 70)=630 .

Example.

What is LCM(68, 34) equal to?

Solution.

Because 68 is divisible by 34, then GCD(68, 34)=34. Now we calculate the least common multiple: GCD(68, 34)=68·34:GCD(68, 34)= 68·34:34=68.

Answer:

LCM(68, 34)=68 .

Note that the previous example fits the following rule for finding the LCM for positive integers a and b: if the number a is divisible by b, then the least common multiple of these numbers is a.

Finding LCM by factoring numbers

Another way to find the least common multiple is based on factoring numbers into prime factors. If you compose a product from all the prime factors of given numbers, and then exclude from this product all the common prime factors present in the decompositions of the given numbers, then the resulting product will be equal to the least common multiple of the given numbers.

The stated rule for finding the LCM follows from the equality LCM(a, b)=a b:GCD(a, b). Indeed, the product of numbers a and b is equal to the product of all factors involved in the expansion of numbers a and b. In turn, GCD(a, b) is equal to the product of all prime factors simultaneously present in the expansions of numbers a and b (as described in the section on finding GCD using the expansion of numbers into prime factors).

Let's give an example. Let us know that 75=3·5·5 and 210=2·3·5·7. Let's compose the product from all the factors of these expansions: 2·3·3·5·5·5·7 . Now from this product we exclude all the factors present in both the expansion of the number 75 and the expansion of the number 210 (these factors are 3 and 5), then the product will take the form 2·3·5·5·7. The value of this product is equal to the least common multiple of 75 and 210, that is, NOC(75, 210)= 2·3·5·5·7=1,050.

Example.

Factor the numbers 441 and 700 into prime factors and find the least common multiple of these numbers.

Solution.

Let's factor the numbers 441 and 700 into prime factors:

We get 441=3·3·7·7 and 700=2·2·5·5·7.

Now let’s create a product from all the factors involved in the expansion of these numbers: 2·2·3·3·5·5·7·7·7. Let us exclude from this product all factors that are simultaneously present in both expansions (there is only one such factor - this is the number 7): 2·2·3·3·5·5·7·7. Thus, LCM(441, 700)=2·2·3·3·5·5·7·7=44 100.

Answer:

NOC(441, 700)= 44 100 .

The rule for finding the LCM using factorization of numbers into prime factors can be formulated a little differently. If the missing factors from the expansion of number b are added to the factors from the expansion of the number a, then the value of the resulting product will be equal to the least common multiple of the numbers a and b.

For example, let's take the same numbers 75 and 210, their decompositions into prime factors are as follows: 75=3·5·5 and 210=2·3·5·7. To the factors 3, 5 and 5 from the expansion of the number 75 we add the missing factors 2 and 7 from the expansion of the number 210, we obtain the product 2·3·5·5·7, the value of which is equal to LCM(75, 210).

Example.

Find the least common multiple of 84 and 648.

Solution.

We first obtain the decompositions of the numbers 84 and 648 into prime factors. They look like 84=2·2·3·7 and 648=2·2·2·3·3·3·3. To the factors 2, 2, 3 and 7 from the expansion of the number 84 we add the missing factors 2, 3, 3 and 3 from the expansion of the number 648, we obtain the product 2 2 2 3 3 3 3 7, which is equal to 4 536 . Thus, the desired least common multiple of 84 and 648 is 4,536.

Answer:

LCM(84, 648)=4,536 .

Finding the LCM of three or more numbers

The least common multiple of three or more numbers can be found by sequentially finding the LCM of two numbers. Let us recall the corresponding theorem, which gives a way to find the LCM of three or more numbers.

Theorem.

Let positive integer numbers a 1 , a 2 , …, a k be given, the least common multiple m k of these numbers is found by sequentially calculating m 2 = LCM(a 1 , a 2) , m 3 = LCM(m 2 , a 3) , … , m k = LCM(m k−1 , a k) .

Let's consider the application of this theorem using the example of finding the least common multiple of four numbers.

Example.

Find the LCM of four numbers 140, 9, 54 and 250.

Solution.

In this example, a 1 =140, a 2 =9, a 3 =54, a 4 =250.

First we find m 2 = LOC(a 1 , a 2) = LOC(140, 9). To do this, using the Euclidean algorithm, we determine GCD(140, 9), we have 140=9·15+5, 9=5·1+4, 5=4·1+1, 4=1·4, therefore, GCD(140, 9)=1 , from where GCD(140, 9)=140 9:GCD(140, 9)= 140·9:1=1,260. That is, m 2 =1 260.

Now we find m 3 = LOC (m 2 , a 3) = LOC (1 260, 54). Let's calculate it through GCD(1 260, 54), which we also determine using the Euclidean algorithm: 1 260=54·23+18, 54=18·3. Then gcd(1,260, 54)=18, from which gcd(1,260, 54)= 1,260·54:gcd(1,260, 54)= 1,260·54:18=3,780. That is, m 3 =3 780.

All that remains is to find m 4 = LOC(m 3, a 4) = LOC(3 780, 250). To do this, we find GCD(3,780, 250) using the Euclidean algorithm: 3,780=250·15+30, 250=30·8+10, 30=10·3. Therefore, GCM(3,780, 250)=10, whence GCM(3,780, 250)= 3 780 250: GCD(3 780, 250)= 3,780·250:10=94,500. That is, m 4 =94,500.

So the least common multiple of the original four numbers is 94,500.

Answer:

LCM(140, 9, 54, 250)=94,500.

In many cases, it is convenient to find the least common multiple of three or more numbers using prime factorizations of the given numbers. In this case, you should adhere to the following rule. The least common multiple of several numbers is equal to the product, which is composed as follows: the missing factors from the expansion of the second number are added to all factors from the expansion of the first number, the missing factors from the expansion of the third number are added to the resulting factors, and so on.

Let's look at an example of finding the least common multiple using prime factorization.

Example.

Find the least common multiple of the five numbers 84, 6, 48, 7, 143.

Solution.

First, we obtain decompositions of these numbers into prime factors: 84=2·2·3·7, 6=2·3, 48=2·2·2·2·3, 7 (7 is a prime number, it coincides with its decomposition into prime factors) and 143=11·13.

To find the LCM of these numbers, to the factors of the first number 84 (they are 2, 2, 3 and 7), you need to add the missing factors from the expansion of the second number 6. The decomposition of the number 6 does not contain missing factors, since both 2 and 3 are already present in the decomposition of the first number 84. Next, to the factors 2, 2, 3 and 7 we add the missing factors 2 and 2 from the expansion of the third number 48, we get a set of factors 2, 2, 2, 2, 3 and 7. There will be no need to add multipliers to this set in the next step, since 7 is already contained in it. Finally, to the factors 2, 2, 2, 2, 3 and 7 we add the missing factors 11 and 13 from the expansion of the number 143. We get the product 2·2·2·2·3·7·11·13, which is equal to 48,048.

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 children 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 factors written down.

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

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 a limited number of divisors of a particular number, but there are an infinite number of multiples.

A common multiple of natural numbers is a number that is divisible by them without leaving 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 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.

The decomposition of each number may contain a different number of factors.

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

In the expansion of the smaller number, you should highlight the factors that are missing in the expansion of the first largest 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

Thus, the product of the prime factors of the larger number 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 it is necessary to find the least common multiple of coprime numbers that do not have identical divisors, then their LCM will be equal to their product.

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