Finding the greatest common divisor online. Finding the least common multiple: methods, examples of finding the LCM

The online calculator allows you to quickly find the greatest common divisor and least common multiple of two or any other number of numbers.

Calculator for finding GCD and NOC

Find GCD and NOC

GCD and NOC found: 5806

How to use the calculator

• Enter numbers in the input field
• In case of entering incorrect characters, the input field will be highlighted in red
• press the button "Find GCD and NOC"

How to enter numbers

• Numbers are entered separated by spaces, dots or commas
• The length of the entered numbers is not limited, so finding the gcd and lcm of long numbers will not be difficult

What is NOD and NOK?

Greatest Common Divisor of several numbers is the largest natural integer by which all the original numbers are divisible without a remainder. The greatest common divisor is abbreviated as GCD.
Least common multiple several numbers is the smallest number that is divisible by each of the original numbers without a remainder. The least common multiple is abbreviated as NOC.

How to check if a number is divisible by another number without a remainder?

To find out if one number is divisible by another without a remainder, you can use some properties of divisibility of numbers. Then, by combining them, one can check the divisibility by some of them and their combinations.

Some signs of divisibility of numbers

1. Sign of divisibility of a number by 2
To determine whether a number is divisible by two (whether it is even), it is enough to look at the last digit of this number: if it is equal to 0, 2, 4, 6 or 8, then the number is even, which means it is divisible by 2.
Example: determine if the number 34938 is divisible by 2.
Solution: look at the last digit: 8 means the number is divisible by two.

2. Sign of divisibility of a number by 3
A number is divisible by 3 when the sum of its digits is divisible by 3. Thus, to determine whether a number is divisible by 3, you need to calculate the sum of the digits and check if it is divisible by 3. Even if the sum of the digits turned out to be very large, you can repeat the same process again.
Example: determine if the number 34938 is divisible by 3.
Solution: we count the sum of the digits: 3+4+9+3+8 = 27. 27 is divisible by 3, which means that the number is divisible by three.

3. Sign of divisibility of a number by 5
A number is divisible by 5 when its last digit is zero or five.
Example: determine if the number 34938 is divisible by 5.
Solution: look at the last digit: 8 means the number is NOT divisible by five.

4. Sign of divisibility of a number by 9
This sign is very similar to the sign of divisibility by three: a number is divisible by 9 when the sum of its digits is divisible by 9.
Example: determine if the number 34938 is divisible by 9.
Solution: we calculate the sum of the digits: 3+4+9+3+8 = 27. 27 is divisible by 9, which means that the number is divisible by nine.

How to find GCD and LCM of two numbers

How to find the GCD of two numbers

The simplest way to calculate the greatest common divisor of two numbers is to find all possible divisors of these numbers and choose the largest of them.

Consider this method using the example of finding GCD(28, 36) :

1. We factorize both numbers: 28 = 1 2 2 7 , 36 = 1 2 2 3 3
2. We find common factors, that is, those that both numbers have: 1, 2 and 2.
3. We calculate the product of these factors: 1 2 2 \u003d 4 - this is the greatest common divisor of the numbers 28 and 36.

How to find the LCM of two numbers

There are two most common ways to find the smallest multiple of two numbers. The first way is that you can write out the first multiples of two numbers, and then choose among them such a number that will be common to both numbers and at the same time the smallest. And the second is to find the GCD of these numbers. Let's just consider it.

To calculate the LCM, you need to calculate the product of the original numbers and then divide it by the previously found GCD. Let's find the LCM for the same numbers 28 and 36:

1. Find the product of the numbers 28 and 36: 28 36 = 1008
2. gcd(28, 36) is already known to be 4
3. LCM(28, 36) = 1008 / 4 = 252 .

Finding GCD and LCM for Multiple Numbers

The greatest common divisor can be found for several numbers, and not just for two. For this, the numbers to be found for the greatest common divisor are decomposed into prime factors, then the product of the common prime factors of these numbers is found. Also, to find the GCD of several numbers, you can use the following relation: gcd(a, b, c) = gcd(gcd(a, b), c).

A similar relation also applies to the least common multiple of numbers: LCM(a, b, c) = LCM(LCM(a, b), c)

Example: find GCD and LCM for numbers 12, 32 and 36.

1. First, let's factorize the numbers: 12 = 1 2 2 3 , 32 = 1 2 2 2 2 2 , 36 = 1 2 2 3 3 .
2. Let's find common factors: 1, 2 and 2 .
3. Their product will give gcd: 1 2 2 = 4
4. Now let's find the LCM: for this we first find the LCM(12, 32): 12 32 / 4 = 96 .
5. To find the LCM of all three numbers, you need to find the GCD(96, 36): 96 = 1 2 2 2 2 2 3 , 36 = 1 2 2 3 3 , GCD = 1 2 . 2 3 = 12 .
6. LCM(12, 32, 36) = 96 36 / 12 = 288 .

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 factor 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.

Mathematical expressions and tasks require a lot of additional knowledge. NOC is one of the main ones, especially often used in the topic. The topic is studied in high school, while it is not particularly difficult to understand material, it will not be difficult for a person familiar with powers and the multiplication table to select the necessary numbers and find the result.

Definition

A common multiple is a number that can be completely divided into two numbers at the same time (a and b). Most often, this number is obtained by multiplying the original numbers a and b. The number must be divisible by both numbers at once, without deviations.

NOC is a short name, which is taken from the first letters.

Ways to get a number

To find the LCM, the method of multiplying numbers is not always suitable, it is much better suited for simple one-digit or two-digit numbers. It is customary to divide into factors, the larger the number, the more factors there will be.

Example #1

For the simplest example, schools usually take simple, one-digit or two-digit numbers. For example, you need to solve the following task, find the least common multiple of the numbers 7 and 3, the solution is quite simple, just multiply them. As a result, there is the number 21, there is simply no smaller number.

Example #2

The second option is much more difficult. The numbers 300 and 1260 are given, finding the LCM is mandatory. To solve the task, the following actions are assumed:

Decomposition of the first and second numbers into the simplest factors. 300 = 2 2 * 3 * 5 2 ; 1260 = 2 2 * 3 2 * 5 * 7. The first stage has been completed.

The second stage involves working with the already obtained data. Each of the received numbers must participate in the calculation of the final result. For each factor, the largest number of occurrences is taken from the original numbers. LCM is a common number, so the factors from the numbers must be repeated in it to the last, even those that are present in one instance. Both initial numbers have in their composition the numbers 2, 3 and 5, in different degrees, 7 is only in one case.

To calculate the final result, you need to take each number in the largest of their represented powers, into the equation. It remains only to multiply and get the answer, with the correct filling, the task fits into two steps without explanation:

1) 300 = 2 2 * 3 * 5 2 ; 1260 = 2 2 * 3 2 *5 *7.

2) NOK = 6300.

That's the whole task, if you try to calculate the desired number by multiplying, then the answer will definitely not be correct, since 300 * 1260 = 378,000.

Examination:

6300 / 300 = 21 - true;

6300 / 1260 = 5 is correct.

The correctness of the result is determined by checking - dividing the LCM by both original numbers, if the number is an integer in both cases, then the answer is correct.

What does NOC mean in mathematics

As you know, there is not a single useless function in mathematics, this one is no exception. The most common purpose of this number is to bring fractions to a common denominator. What is usually studied in grades 5-6 of high school. It is also additionally a common divisor for all multiples, if such conditions are in the problem. Such an expression can find a multiple not only of two numbers, but also of a much larger number - three, five, and so on. The more numbers - the more actions in the task, but the complexity of this does not increase.

For example, given the numbers 250, 600 and 1500, you need to find their total LCM:

1) 250 = 25 * 10 = 5 2 * 5 * 2 = 5 3 * 2 - this example describes the factorization in detail, without reduction.

2) 600 = 60 * 10 = 3 * 2 3 *5 2 ;

3) 1500 = 15 * 100 = 33 * 5 3 *2 2 ;

In order to compose an expression, it is required to mention all the factors, in this case 2, 5, 3 are given - for all these numbers it is required to determine the maximum degree.

Attention: all multipliers must be brought to full simplification, if possible, decomposing to the level of single digits.

Examination:

1) 3000 / 250 = 12 - true;

2) 3000 / 600 = 5 - true;

3) 3000 / 1500 = 2 is correct.

This method does not require any tricks or genius level abilities, everything is simple and clear.

Another way

In mathematics, a lot is connected, a lot can be solved in two or more ways, the same goes for finding the least common multiple, LCM. The following method can be used in the case of simple two-digit and single-digit numbers. A table is compiled in which the multiplier is entered vertically, the multiplier horizontally, and the product is indicated in the intersecting cells of the column. You can reflect the table by means of a line, a number is taken and the results of multiplying this number by integers are written in a row, from 1 to infinity, sometimes 3-5 points are enough, the second and subsequent numbers are subjected to the same computational process. Everything happens until a common multiple is found.

Given the numbers 30, 35, 42, you need to find the LCM that connects all the numbers:

1) Multiples of 30: 60, 90, 120, 150, 180, 210, 250, etc.

2) Multiples of 35: 70, 105, 140, 175, 210, 245, etc.

3) Multiples of 42: 84, 126, 168, 210, 252, etc.

It is noticeable that all the numbers are quite different, the only common number among them is 210, so it will be the LCM. Among the processes associated with this calculation, there is also the greatest common divisor, which is calculated according to similar principles and is often encountered in neighboring problems. The difference is small, but significant enough, LCM involves the calculation of a number that is divisible by all given initial values, and GCD assumes the calculation of the largest value by which the initial numbers are divided.

A multiple of a number 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 evenly divisible by each number in the group. To find the least common multiple, you need to find the prime factors of the given numbers. Also, LCM can be calculated using a number of other methods that are applicable to groups of two or more numbers.

Steps

A number of multiples

Look at these numbers. The method described here is best used when given two numbers that are both less than 10. If large numbers are given, use a different method.

• For example, find the least common multiple of the numbers 5 and 8. These are small numbers, so this method can be used.

1. A multiple of a number is a number that is divisible by a given number without a remainder. Multiple numbers 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 rows 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 appears in both series of multiples. You may have to write long series of multiples to find the total. The smallest number that appears in both series of multiples is the least common multiple.

   • For example, the smallest number that appears in the series of multiples of 5 and 8 is 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 that are both 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 this method can be used.

   2. Factorize the first number. That is, you need to find such prime numbers, when multiplied, you get a given number. Having found prime factors, write them down as an equality.

      • 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 down 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 get this 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 down as an expression: .

   4. Write down the factors common to both numbers. Write such factors as a multiplication operation. As you write down each factor, cross it out in both expressions (expressions that describe the decomposition of numbers into prime factors).

      • For example, the common factor for both numbers is 2, so write 2 × (\displaystyle 2\times ) and cross out the 2 in both expressions.
      • The common factor for both numbers 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 as follows: 2 × 2 × 5 (\displaystyle 2\times 2\times 5)
      • In the expression 84 = 2 × 7 × 3 × 2 (\displaystyle 84=2\times 7\times 3\times 2) both deuces (2) are also crossed out. Factors 7 and 3 are not crossed out, so write the multiplication operation as follows: 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 divisors

   1. Draw a grid like you would for a game of tic-tac-toe. Such a grid consists of two parallel lines that intersect (at right angles) with two other parallel lines. This will result in three rows and three columns (the grid looks a lot like the # sign). Write the first number in the first row and second column. Write the second number in the first row and third column.

      • For example, find the least common multiple of 18 and 30. Write 18 in the first row and second column, and write 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 divisors, but this is not a prerequisite.

      • For example, 18 and 30 are even numbers, so their common divisor 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 corresponding number. The 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 15 under 30.

   4. Find a divisor common to both quotients. If there is no such divisor, skip the next two steps. Otherwise, write down 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 the 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, supplement the grid with additional cells. Repeat the above 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 highlighted 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 the 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 by which to divide. The quotient is the result of dividing two numbers. The remainder is the number left when two numbers are divided.

      • For example, in the expression 15 ÷ 6 = 2 (\displaystyle 15\div 6=2) rest. 3:
        15 is the divisible
        6 is the divisor
        2 is private
        3 is the remainder.

The material presented below is a logical continuation of the theory from the article under the heading LCM - least common multiple, definition, examples, relationship between LCM and GCD. Here we will talk about finding the least common multiple (LCM), and pay special attention to solving examples. Let us first show how the LCM of two numbers is calculated in terms of the GCD of these numbers. Next, consider finding the least common multiple by factoring numbers into prime factors. After that, we will focus on finding the LCM of three or more numbers, and also pay attention to the calculation of the LCM of negative numbers.

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Calculation of the least common multiple (LCM) through gcd

One way to find the least common multiple is based on the relationship between LCM and GCD. The existing relationship between LCM and GCD allows you to calculate the least common multiple of two positive integers through the known greatest common divisor. The corresponding formula has the form LCM(a, b)=a b: GCM(a, b) . Consider examples of finding the LCM according to the above formula.

Example.

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

Solution.

In this example a=126 , b=70 . Let us use the relationship between LCM and GCD expressed by the formula LCM(a, b)=a b: GCM(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 according to the written formula.

Find gcd(126, 70) using Euclid's algorithm: 126=70 1+56 , 70=56 1+14 , 56=14 4 , hence gcd(126, 70)=14 .

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

Answer:

LCM(126, 70)=630 .

Example.

What is LCM(68, 34) ?

Solution.

Because 68 is evenly divisible by 34 , then gcd(68, 34)=34 . Now we calculate the least common multiple: LCM(68, 34)=68 34: LCM(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 the LCM by Factoring Numbers into Prime Factors

Another way to find the least common multiple is based on factoring numbers into prime factors. If we make a product of all prime factors of these numbers, after which we exclude from this product all common prime factors that are present in the expansions of these numbers, then the resulting product will be equal to the least common multiple of these numbers.

The announced rule for finding the LCM follows from the equality LCM(a, b)=a b: GCM(a, b). Indeed, the product of the numbers a and b is equal to the product of all the factors involved in the expansions of the numbers a and b. In turn, gcd(a, b) is equal to the product of all prime factors that are simultaneously present in the expansions of the numbers a and b (which is described in the section on finding the gcd using the decomposition of numbers into prime factors).

Let's take an example. Let we know that 75=3 5 5 and 210=2 3 5 7 . Compose the product of all factors of these expansions: 2 3 3 5 5 5 7 . Now we exclude from this product all the factors that are present both in the expansion of the number 75 and in the expansion of the number 210 (such 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 the numbers 75 and 210, that is, LCM(75, 210)= 2 3 5 5 7=1 050.

Example.

After factoring the numbers 441 and 700 into prime factors, find the least common multiple of these numbers.

Solution.

Let's decompose 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 make a product of all the factors involved in the expansions of these numbers: 2 2 3 3 5 5 7 7 7 . Let us exclude from this product all the 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 . In this way, LCM(441, 700)=2 2 3 3 5 5 7 7=44 100.

Answer:

LCM(441, 700)= 44 100 .

The rule for finding the LCM using the decomposition of numbers into prime factors can be formulated a little differently. If we add the missing factors from the expansion of the number b 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 all the same numbers 75 and 210, their expansions 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 decomposition of the number 75, we add the missing factors 2 and 7 from the decomposition of the number 210, we get the product 2 3 5 5 7 , the value of which is LCM(75, 210) .

Example.

Find the least common multiple of 84 and 648.

Solution.

We first obtain the decomposition 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 decomposition of the number 84 we add the missing factors 2 , 3 , 3 and 3 from the decomposition of the number 648 , we get the product 2 2 2 3 3 3 3 7 , which is equal to 4 536 . Thus, the desired least common multiple of the numbers 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 successively finding the LCM of two numbers. Recall the corresponding theorem, which gives a way to find the LCM of three or more numbers.

Theorem.

Let positive integers a 1 , a 2 , …, a k be given, the least common multiple m k of these numbers is found in the sequential calculation m 2 = LCM (a 1 , a 2) , m 3 = LCM (m 2 , a 3) , … , m k =LCM(m k−1 , a k) .

Consider the application of this theorem on the example of finding the least common multiple of four numbers.

Example.

Find the LCM of the 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 \u003d LCM (a 1, a 2) \u003d LCM (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 , whence LCM(140, 9)=140 9: LCM(140, 9)= 140 9:1=1 260 . That is, m 2 =1 260 .

Now we find m 3 \u003d LCM (m 2, a 3) \u003d LCM (1 260, 54). Let's calculate it through gcd(1 260, 54) , which is also determined by the Euclid algorithm: 1 260=54 23+18 , 54=18 3 . Then gcd(1 260, 54)=18 , whence LCM(1 260, 54)= 1 260 54:gcd(1 260, 54)= 1 260 54:18=3 780 . That is, m 3 \u003d 3 780.

Left to find m 4 \u003d LCM (m 3, a 4) \u003d LCM (3 780, 250). To do this, we find GCD(3 780, 250) using the Euclid algorithm: 3 780=250 15+30 , 250=30 8+10 , 30=10 3 . Therefore, gcd(3 780, 250)=10 , whence gcd(3 780, 250)= 3 780 250:gcd(3 780, 250)= 3 780 250:10=94 500 . That is, m 4 \u003d 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, the least common multiple of three or more numbers is conveniently found using prime factorizations of given numbers. In this case, the following rule should be followed. 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 the factors from the expansion of the first number, the missing factors from the expansion of the third number are added to the obtained factors, and so on.

Consider an example of finding the least common multiple using the decomposition of numbers into prime factors.

Example.

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

Solution.

First, we obtain the expansions of these numbers into prime factors: 84=2 2 3 7 , 6=2 3 , 48=2 2 2 2 3 , 7 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 expansion of the number 6 does not contain missing factors, since both 2 and 3 are already present in the expansion of the first number 84 . Further 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 is no need to add factors 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 .