Greatest natural number, by which the numbers a and b are divided without a remainder, are called greatest common divisor these numbers. Denote GCD(a, b).

Consider finding the GCD using the example of two natural numbers 18 and 60:

324 , 111 and 432

Let's decompose the numbers into prime factors:

324 = 2×2×3×3×3×3

111 = 3×37

432 = 2×2×2×2×3×3×3

Delete from the first number, the factors of which are not in the second and third numbers, we get:

2 x 2 x 2 x 2 x 3 x 3 x 3 = 3

As a result of GCD( 324 , 111 , 432 )=3

Finding GCD with Euclid's Algorithm

The second way to find the greatest common divisor using Euclid's algorithm. Euclid's algorithm is the most effective way finding GCD, using it you need to constantly find the remainder of the division of numbers and apply recurrent formula.

Recurrent formula for GCD, gcd(a, b)=gcd(b, a mod b), where a mod b is the remainder of dividing a by b.

Euclid's algorithm

Example Find the Greatest Common Divisor of Numbers 7920 and 594

Let's find GCD( 7920 , 594 ) using the Euclid algorithm, we will calculate the remainder of the division using a calculator.

• 7920 mod 594 = 7920 - 13 × 594 = 198
• 594 mod 198 = 594 - 3 × 198 = 0

As a result, we get GCD( 7920 , 594 ) = 198

Least common multiple

In order to find common denominator when adding and subtracting fractions with different denominators need to know and be able to calculate least common multiple(NOC).

A multiple of the number "a" is a number that is itself divisible by the number "a" without a remainder.

Numbers that are multiples of 8 (that is, these numbers will be divided by 8 without a remainder): these are the numbers 16, 24, 32 ...

Multiples of 9: 18, 27, 36, 45…

There are infinitely many multiples of a given number a, in contrast to the divisors of the same number. Divisors - a finite number.

A common multiple of two natural numbers is a number that is evenly divisible by both of these numbers..

Least common multiple(LCM) of two or more natural numbers is the smallest natural number that is itself divisible by each of these numbers.

How to find the NOC

LCM can be found and written in two ways.

The first way to find the LCM

This method is usually used for small numbers.

1. We write the multiples for each of the numbers in a line until there is a multiple that is the same for both numbers.
2. A multiple of the number "a" is denoted by a capital letter "K".

Example. Find LCM 6 and 8.

The second way to find the LCM

This method is convenient to use to find the LCM for three or more numbers.

The number of identical factors in the expansions of numbers can be different.

You can also formalize finding the least common multiple (LCM) as follows. Let's find the LCM (12, 16, 24) .

24 = 2 2 2 3

As we can see from the expansion of numbers, all factors of 12 are included in the expansion of 24 (the largest of the numbers), so we add only one 2 from the expansion of the number 16 to the LCM.

LCM (12, 16, 24) = 2 2 2 3 2 = 48

Answer: LCM (12, 16, 24) = 48

Special cases of finding NOCs

For example, LCM(60, 15) = 60
Since coprime numbers have no common prime divisors, their least common multiple is equal to the product of these numbers.

On our site, you can also use a special calculator to find the least common multiple online to check your calculations.

If a natural number is only divisible by 1 and itself, then it is called prime.

Any natural number is always divisible by 1 and itself.

The number 2 is the smallest prime number. This is the only even prime number, the rest of the prime numbers are odd.

There are many prime numbers, and the first among them is the number 2. However, there is no last prime number. In the "For Study" section, you can download a table of prime numbers up to 997.

But many natural numbers are evenly divisible by other natural numbers.

• the number 12 is divisible by 1, by 2, by 3, by 4, by 6, by 12;
• 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 evenly divisible (for 12 these are 1, 2, 3, 4, 6 and 12) are called the divisors of the number.

The divisor of a natural number a is such a natural number that divides the given number "a" without a remainder.

A natural number that has more than two factors is called a composite number.

Note that the numbers 12 and 36 have common divisors. These are numbers: 1, 2, 3, 4, 6, 12. The largest divisor of these numbers is 12.

The common divisor of two given numbers "a" and "b" is the number by which both given numbers "a" and "b" are divided without remainder.

Greatest Common Divisor(gcd) of two given numbers "a" and "b" is largest number, by which both numbers "a" and "b" are divisible without a remainder.

Briefly, the greatest common divisor of numbers "a" and "b" is written as follows:

Example: gcd (12; 36) = 12 .

The divisors of numbers in the solution record are denoted by a capital letter "D".

The numbers 7 and 9 have only one common divisor - the number 1. Such numbers are called coprime numbers.

Coprime numbers are natural numbers that have only one common divisor - the number 1. Their GCD is 1.

How to find the greatest common divisor

To find the gcd of two or more natural numbers you need:

• decompose the divisors of numbers into prime factors;

Calculations are conveniently written using a vertical bar. To the left of the line, first write down the dividend, to the right - the divisor. Further in the left column we write down the values ​​of private.

Let's explain right away with an example. Let's factorize the numbers 28 and 64 into prime factors.

Underline the same prime factors in both numbers.
28 = 2 2 7

64 = 2 2 2 2 2 2
We find the product of identical prime factors and write down the answer;
GCD (28; 64) = 2 2 = 4

Answer: GCD (28; 64) = 4

You can arrange the location of the GCD in two ways: in a column (as was done above) or “in a line”.

The first way to write GCD

Find GCD 48 and 36.

GCD (48; 36) = 2 2 3 = 12

The second way to write GCD

Now let's write the GCD search solution in a line. Find GCD 10 and 15.

On our information site, you can also find the greatest common divisor online using the helper program to check your calculations.

Finding the least common multiple, methods, examples of finding the LCM.

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: GCD(a, b). Consider examples of finding the LCM according to the above formula.

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

In this example a=126 , b=70 . Let's use the link of LCM with GCD, which is 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:GCD(126, 70)= 126 70:14=630 .

What is LCM(68, 34) ?

Since 68 is evenly divisible by 34 , then gcd(68, 34)=34 . Now we calculate the least common multiple: LCM(68, 34)=68 34:GCD(68, 34)= 68 34:34=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 75 and 210 , that is, LCM(75, 210)= 2 3 5 5 7=1 050 .

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

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 . So LCM(441, 700)=2 2 3 3 5 5 7 7=44 100 .

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) .

Find the least common multiple of 84 and 648.

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.

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.

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

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

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

First we find m 2 = LCM (a 1 , a 2) = 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: GCD(140, 9)= 140 9:1=1 260 . That is, m 2 =1 260 .

Now we find m 3 = LCM (m 2 , a 3) = 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.

It remains to find m 4 = LCM (m 3 , a 4) = 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 , hence LCM(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.

LCM(140, 9, 54, 250)=94500 .

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.

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

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 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 .

Therefore, LCM(84, 6, 48, 7, 143)=48048 .

LCM(84, 6, 48, 7, 143)=48048 .

Finding the Least Common Multiple of Negative Numbers

Sometimes there are tasks in which you need to find the least common multiple of numbers, among which one, several or all numbers are negative. In these cases, all negative numbers must be replaced by their opposite numbers, after which the LCM of positive numbers should be found. This is the way to find the LCM of negative numbers. For example, LCM(54, −34)=LCM(54, 34) and LCM(−622, −46, −54, −888)= LCM(622, 46, 54, 888) .

We can do this because the set of multiples of a is the same as the set of multiples of −a (a and −a are opposite numbers). Indeed, let b be some multiple of a , then b is divisible by a , and the concept of divisibility asserts the existence of such an integer q that b=a q . But the equality b=(−a)·(−q) will also be true, which, by virtue of the same concept of divisibility, means that b is divisible by −a , that is, b is a multiple of −a . The converse statement is also true: if b is some multiple of −a , then b is also a multiple of a .

Find the least common multiple of the negative numbers −145 and −45.

Let's replace the negative numbers −145 and −45 with their opposite numbers 145 and 45 . We have LCM(−145, −45)=LCM(145, 45) . Having determined gcd(145, 45)=5 (for example, using the Euclid algorithm), we calculate LCM(145, 45)=145 45:gcd(145, 45)= 145 45:5=1 305 . Thus, the least common multiple of the negative integers −145 and −45 is 1,305 .

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We continue to study division. In this lesson, we will look at concepts such as GCD and NOC.

GCD is the greatest common divisor.

NOC is the least common multiple.

The topic is rather boring, but it is necessary to understand it. Without understanding this topic, you will not be able to work effectively with fractions, which are a real obstacle in mathematics.

Greatest Common Divisor

Definition. Greatest Common Divisor of Numbers a and b a and b divided without remainder.

In order to understand this definition well, we substitute instead of variables a and b any two numbers, for example, instead of a variable a substitute the number 12, and instead of the variable b number 9. Now let's try to read this definition:

Greatest Common Divisor of Numbers 12 and 9 is the largest number by which 12 and 9 divided without remainder.

It is clear from the definition that we are talking about a common divisor of the numbers 12 and 9, and this divisor is the largest of all existing divisors. This greatest common divisor (gcd) must be found.

To find the greatest common divisor of two numbers, three methods are used. The first method is quite time-consuming, but it allows you to understand the essence of the topic well and feel its whole meaning.

The second and third methods are quite simple and make it possible to quickly find the GCD. We will consider all three methods. And what to apply in practice - you choose.

The first way is to find all possible divisors of two numbers and choose the largest of them. Let's consider this method in the following example: find the greatest common divisor of the numbers 12 and 9.

First, we find all possible divisors of the number 12. To do this, we divide 12 into all divisors in the range from 1 to 12. If the divisor allows us to divide 12 without a remainder, then we will highlight it in blue and make an appropriate explanation in brackets.

12: 1 = 12
(12 divided by 1 without a remainder, so 1 is a divisor of 12)

12: 2 = 6
(12 divided by 2 without a remainder, so 2 is a divisor of 12)

12: 3 = 4
(12 divided by 3 without a remainder, so 3 is a divisor of 12)

12: 4 = 3
(12 divided by 4 without a remainder, so 4 is a divisor of 12)

12:5 = 2 (2 left)
(12 is not divided by 5 without a remainder, so 5 is not a divisor of 12)

12: 6 = 2
(12 divided by 6 without a remainder, so 6 is a divisor of 12)

12: 7 = 1 (5 left)
(12 is not divided by 7 without a remainder, so 7 is not a divisor of 12)

12: 8 = 1 (4 left)
(12 is not divided by 8 without a remainder, so 8 is not a divisor of 12)

12:9 = 1 (3 left)
(12 is not divided by 9 without a remainder, so 9 is not a divisor of 12)

12: 10 = 1 (2 left)
(12 is not divided by 10 without a remainder, so 10 is not a divisor of 12)

12:11 = 1 (1 left)
(12 is not divided by 11 without a remainder, so 11 is not a divisor of 12)

12: 12 = 1
(12 divided by 12 without a remainder, so 12 is a divisor of 12)

Now let's find the divisors of the number 9. To do this, check all the divisors from 1 to 9

9: 1 = 9
(9 divided by 1 without a remainder, so 1 is a divisor of 9)

9: 2 = 4 (1 left)
(9 is not divided by 2 without a remainder, so 2 is not a divisor of 9)

9: 3 = 3
(9 divided by 3 without a remainder, so 3 is a divisor of 9)

9: 4 = 2 (1 left)
(9 is not divided by 4 without a remainder, so 4 is not a divisor of 9)

9:5 = 1 (4 left)
(9 is not divided by 5 without a remainder, so 5 is not a divisor of 9)

9: 6 = 1 (3 left)
(9 did not divide by 6 without a remainder, so 6 is not a divisor of 9)

9:7 = 1 (2 left)
(9 is not divided by 7 without a remainder, so 7 is not a divisor of 9)

9:8 = 1 (1 left)
(9 is not divided by 8 without a remainder, so 8 is not a divisor of 9)

9: 9 = 1
(9 divided by 9 without a remainder, so 9 is a divisor of 9)

Now write down the divisors of both numbers. The numbers highlighted in blue are the divisors. Let's write them out:

Having written out the divisors, you can immediately determine which one is the largest and most common.

By definition, the greatest common divisor of 12 and 9 is the number by which 12 and 9 are evenly divisible. The greatest and common divisor of the numbers 12 and 9 is the number 3

Both the number 12 and the number 9 are divisible by 3 without a remainder:

So gcd (12 and 9) = 3

The second way to find GCD

Now consider the second way to find the greatest common divisor. The essence of this method is to decompose both numbers into prime factors and multiply the common ones.

Example 1. Find GCD of numbers 24 and 18

First, let's factor both numbers into prime factors:

Now let's multiply them common factors. In order not to get confused, the common factors can be underlined.

We look at the decomposition of the number 24. Its first factor is 2. We are looking for the same factor in the decomposition of the number 18 and see that it is also there. We underline both twos:

Again we look at the decomposition of the number 24. Its second factor is also 2. We are looking for the same factor in the decomposition of the number 18 and see that it is not there for the second time. Then we don't highlight anything.

The next two in the expansion of the number 24 is also missing in the expansion of the number 18.

We pass to the last factor in the decomposition of the number 24. This is the factor 3. We are looking for the same factor in the decomposition of the number 18 and see that it is also there. We emphasize both threes:

So, the common factors of the numbers 24 and 18 are the factors 2 and 3. To get the GCD, these factors must be multiplied:

So gcd (24 and 18) = 6

The third way to find GCD

Now consider the third way to find the greatest common divisor. The essence of this method lies in the fact that the numbers to be searched for the greatest common divisor are decomposed into prime factors. Then, from the decomposition of the first number, factors that are not included in the decomposition of the second number are deleted. The remaining numbers in the first expansion are multiplied and get GCD.

For example, let's find the GCD for the numbers 28 and 16 in this way. First of all, we decompose these numbers into prime factors:

We got two expansions: and

Now, from the expansion of the first number, we delete the factors that are not included in the expansion of the second number. The expansion of the second number does not include seven. We will delete it from the first expansion:

Now we multiply the remaining factors and get the GCD:

The number 4 is the greatest common divisor of the numbers 28 and 16. Both of these numbers are divisible by 4 without a remainder:

Example 2 Find GCD of numbers 100 and 40

Factoring out the number 100

Factoring out the number 40

We got two expansions:

Now, from the expansion of the first number, we delete the factors that are not included in the expansion of the second number. The expansion of the second number does not include one five (there is only one five). We delete it from the first decomposition

Multiply the remaining numbers:

We got the answer 20. So the number 20 is the greatest common divisor of the numbers 100 and 40. These two numbers are divisible by 20 without a remainder:

GCD (100 and 40) = 20.

Example 3 Find the gcd of the numbers 72 and 128

Factoring out the number 72

Factoring out the number 128

2×2×2×2×2×2×2

Now, from the expansion of the first number, we delete the factors that are not included in the expansion of the second number. The expansion of the second number does not include two triplets (there are none at all). We delete them from the first decomposition:

We got the answer 8. So the number 8 is the greatest common divisor of the numbers 72 and 128. These two numbers are divisible by 8 without a remainder:

GCD (72 and 128) = 8

Finding GCD for Multiple Numbers

The greatest common divisor can be found for several numbers, and not just for two. For this, the numbers to be searched for the greatest common divisor are decomposed into prime factors, then the product of the common prime factors of these numbers is found.

For example, let's find the GCD for the numbers 18, 24 and 36

Factoring the number 18

Factoring the number 24

Factoring the number 36

We got three expansions:

Now we select and underline the common factors in these numbers. Common factors must be included in all three numbers:

We see that the common factors for the numbers 18, 24 and 36 are factors 2 and 3. By multiplying these factors, we get the GCD we are looking for:

We got the answer 6. So the number 6 is the greatest common divisor of the numbers 18, 24 and 36. These three numbers are divisible by 6 without a remainder:

GCD (18, 24 and 36) = 6

Example 2 Find gcd for numbers 12, 24, 36 and 42

Let's factorize each number. Then we find the product of the common factors of these numbers.

Factoring the number 12

Factoring the number 42

We got four expansions:

Now we select and underline the common factors in these numbers. Common factors must be included in all four numbers:

We see that the common factors for the numbers 12, 24, 36, and 42 are the factors 2 and 3. By multiplying these factors, we get the GCD we are looking for:

We got the answer 6. So the number 6 is the greatest common divisor of the numbers 12, 24, 36 and 42. These numbers are divisible by 6 without a remainder:

gcd(12, 24, 36 and 42) = 6

From the previous lesson, we know that if some number is divided by another without a remainder, it is called a multiple of this number.

It turns out that a multiple can be common to several numbers. And now we will be interested in a multiple of two numbers, while it should be as small as possible.

Definition. Least common multiple (LCM) of numbers a and b- a and b a and number b.

Definition contains two variables a and b. Let's substitute any two numbers for these variables. For example, instead of a variable a substitute the number 9, and instead of the variable b let's substitute the number 12. Now let's try to read the definition:

Least common multiple (LCM) of numbers 9 and 12 - is the smallest number that is a multiple of 9 and 12 . In other words, it is such a small number that is divisible without a remainder by the number 9 and on the number 12 .

It is clear from the definition that the LCM is the smallest number that is divisible without a remainder by 9 and 12. This LCM is required to be found.

There are two ways to find the least common multiple (LCM). The first way is that you can write down the first multiples of two numbers, and then choose among these multiples such a number that will be common to both numbers and small. Let's apply this method.

First of all, let's find the first multiples for the number 9. To find the multiples for 9, you need to multiply this nine by the numbers from 1 to 9 in turn. The answers you get will be multiples of the number 9. So, let's start. Multiples will be highlighted in red:

Now we find multiples for the number 12. To do this, we multiply 12 by all the numbers 1 to 12 in turn.

The greatest common divisor is another indicator that makes it easier to work with fractions. Very often, as a result of calculations, fractions are obtained with very large values ​​​​of the numerator and denominator. It is possible to reduce such numbers in stages, but this is extremely long, so it is easier to immediately find the GCD and reduce it. Let's take a closer look at the topic.

What is NOD?

The greatest common divisor (GCD) of a series of numbers is the largest number by which each of the numbers in the series can be divided without a remainder.

How to find NOD?

In order to find the GCD, it is necessary to decompose each of the numbers into prime factors and highlight the common part.

They didn’t come up with a special formula for this, but there is a calculation algorithm.

Let's give an example of finding the greatest common divisor of two natural numbers: 540 and 252. Let's decompose 640 into prime factors. The sequence of actions is as follows:

• We divide the number by the smallest possible prime number. That is, if the number can be divided by 2, 3 or 5, then you must first divide by 5. Just so as not to get confused.
• The resulting result is divided by the smallest possible prime number.
• We repeat the division of each result obtained until we get a prime number.

Now we will carry out the same procedure in practice.

• 540: 2=270
• 270:2=135
• 135: 3 =45
• 45: 3=15
• 15: 5 = 3

Let's write the result as an equation 540=2*2*3*3*3*5. In order to write down the result, you need to multiply the last resulting number by all divisors.

Let's do the same with the number 252:

• 252: 2=126
• 126: 2=63
• 63: 3=21
• 21: 3 = 7

Let's write down the result: 252=2*2*3*3*7.

Each expansion has the same numbers. Let's find them, these are two numbers 2 and two numbers 3. Only 7 and 3 * 5 differ.

In order to find the GCD, you need to multiply the common factors. That is, there will be two twos and two threes in the product.

GCD=2*2*3*3=36

How can it be used?

Task: reduce the fraction $$252\over540$$.

We have already found the GCD for these two numbers, now we will simply use the already calculated value.

Let's reduce the numerator and denominator of the fraction by 36 and get the answer.

$$(252\over540) =(7\over15)$$ - to quickly reduce, just look at the expansion of numbers.

If 540=2*2*3*3*3*5, and GCD=36=2*2*3*3, then 540 = 36*3*5. And if we divide 540 by 36, we get 3*5=15.

Without GCD, we would have to write abbreviations in one long line. In addition, there are cases when it is not clear whether a fraction can be reduced at all. For such situations in mathematics, they came up with the decomposition of numbers into prime factors and GCD.

What have we learned?

We learned what the greatest common divisor of a pair of numbers is, figured out how to use the indicator in practice, solved the problem of finding GCD and using GCD to reduce fractions. We realized that with the use of GCD, it is easier and faster to reduce bulky fractions by finding the GCD for the numerator and denominator.

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A dividend that is divisible by a given divisor without a remainder is also called multiple. For example, 48 is a multiple of 8, 48 is a multiple, and 8 is a divisor.

A number can be a multiple of not one, but several numbers at once, such a number is called common multiple. For example, the number 77 is a common multiple of the numbers: 1, 7, 11, 77.

Another example. The number 3 is a multiple of 12, 15, 24, 27, 30, etc. The number 5 is a multiple of 10, 15, 25, 30, 35, etc. The numbers 3 and 5 have common multiples of 15 and 30.

Finding a common multiple of several numbers is quite simple, you can simply multiply these numbers, as a result, the product of these numbers will be their common multiple.

NOC

Of all the common multiples for given numbers, the least common multiple is of particular interest.

Least common multiple(abbreviated LCM) of several given numbers is the smallest number that is evenly divisible by each of the given numbers.

For example, for three numbers: 3, 5 and 12, the least common multiple is the number 60, since no other number less than 60 is not completely divisible by 3, 5 and 12.

Usually the least common multiple is written like this: LCM ( a, b, ...) = x.

According to this, we write the least common multiple of the numbers 3, 5 and 12:

LCM (3, 5, 12) = 60.

NOC calculator

This calculator will help you find the least common multiple of numbers. Just enter the numbers separated by spaces or commas and click the Calculate LCM button.

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: GCD(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: 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 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: GCD(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 decomposition 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 , …, ak be given, the least common multiple mk of these numbers is found in the sequential calculation m 2 = LCM (a 1 , a 2) , m 3 = LCM (m 2 , a 3) , … , mk =LCM(mk−1 , ak) .

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 .

One of the tasks that cause a problem for modern schoolchildren, who are used to using calculators built into gadgets in place and out of place, is finding the greatest common divisor (GCD) of two or more numbers.

It is impossible to solve any mathematical problem if it is not known what is actually being asked. To do this, you need to know what this or that expression means. used in mathematics.

General concepts and definitions

Need to know:

1. If a certain number can be used to count various objects, for example, nine pillars, sixteen houses, then it is natural. The smallest of them will be one.
2. When a natural number is divisible by another natural number, the smaller number is said to be a divisor of the larger one.
3. If two or more different numbers are divisible by a certain number without a remainder, then they say that the latter will be their common divisor (OD).
4. The largest of the ODs is called the greatest common divisor (GCD).
5. In such a case, when a number has only two natural divisors (itself and one), it is called prime. The smallest among them is the deuce, besides, it is the only even number in their series.
6. If two numbers have a maximum common divisor of one, then they will be coprime.
7. A number with more than two divisors is called a composite number.
8. The process when all prime factors are found, which, when multiplied with each other, will give the initial value in the product in mathematics, is called decomposition into prime factors. Moreover, the same factors in the expansion can occur more than once.

In mathematics, the following notations are accepted:

1. Divisors D (45) = (1; 3; 5; 9; 45).
2. OD (8;18) = (1;2).
3. GCD (8;18) = 2.

Various ways to find GCD

The easiest question to answer how to find NOD when the smaller number is a divisor of the larger one. It will be the greatest common divisor in this case.

For example, GCD (15;45) = 15, GCD (48;24) = 24.

But such cases in mathematics are very rare, therefore, in order to find the GCD, more complex techniques are used, although it is still highly recommended to check this option before starting work.

Method of decomposition into prime factors

If you need to find the GCD of two or more different numbers, it is enough to decompose each of them into simple factors, and then carry out the process of multiplying those of them that are in each of the numbers.

Example 1

Consider how to find GCD 36 and 90:

1. 36 = 1*2*2*3*3;
2. 90 = 1*2*3*3*5;

GCD (36;90) = 1*2*3*3 = 18.

Now let's see how to find the same in case of three numbers, take for example 54; 162; 42.

We already know how to decompose 36, let's deal with the rest:

1. 162 = 1*2*3*3*3*3;
2. 42 = 1*2*3*7;

Thus, GCD (36;162;42) = 1*2*3 = 6.

It should be noted that it is absolutely optional to write the unit in the expansion.

Consider the way how easy it is to factorize, for this, on the left, we will write the number we need, and on the right, we will write simple divisors.

Columns can be separated by either a division sign or a simple vertical bar.

1. 36 / 2 we will continue our division process;
2. 18 / 2 further;
3. 9 / 3 and again;
4. 3 / 3 is now quite elementary;
5. 1 - the result is ready.

The desired 36 \u003d 2 * 2 * 3 * 3.

Euclidean way

This option has been known to mankind since the time of the ancient Greek civilization, it is much simpler, and is attributed to the great mathematician Euclid, although very similar algorithms were used before. This method is to use the following algorithm, we divide the larger number with a remainder by the smaller one. Then we divide our divisor by the remainder and continue to act in this way in a circle until the division is complete. The last value will turn out to be the desired greatest common divisor.

Let's give an example of using this algorithm:

Let's try to find out which GCD for 816 and 252:

1. 816 / 252 = 3 and the remainder is 60. Now we divide 252 by 60;
2. 252 / 60 = 4 the remainder this time will be 12. Let's continue our circular process, divide sixty by twelve;
3. 60 / 12 = 5. Since this time we did not receive any remainder, we have the result ready, twelve will be the value we are looking for.

So, at the end of our process we got NOD (816;252) = 12.

Actions if it is necessary to determine the GCD if more than two values ​​are specified

We have already figured out what to do in the case when there are two different numbers, now we will learn how to act if there are any. 3 or more.

Despite the seeming complexity, this task will not cause us any problems. Now we choose any two numbers and determine the value we are looking for for them. The next step is to find the GCD for the result obtained and the third of the given values. Then again we act according to the principle already known to us for the fourth fifth and so on.

Conclusion

So, with the seeming great complexity of the task set before us initially, in fact, everything is simple, the main thing is to be able to perform the process of division without error and stick to any of the two algorithms described above.

Although both methods are quite acceptable, in a comprehensive school the first method is much more commonly used.. This is due to the fact that prime factorization will be needed when studying the following learning topic- determination of the greatest common multiple (LCM). But still, it is worth noting again - the use of the Euclid algorithm can in no way be considered erroneous.

Video

With the help of the video, you can learn how to find the greatest common divisor.