Node and nok division online. Nod and nok of numbers - the greatest common divisor and least common multiple of several numbers

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Let's continue the discussion about the least common multiple that we started in the LCM - Least Common Multiple, Definition, Examples section. In this topic, we will look at ways to find the LCM for three numbers or more, we will analyze the question of how to find the LCM of a negative number.

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Calculation of the least common multiple (LCM) through gcd

We have already established the relationship between the least common multiple and the greatest common divisor. Now let's learn how to define the LCM through the GCD. First, let's figure out how to do this for positive numbers.

Definition 1

You can find the least common multiple through the greatest common divisor using the formula LCM (a, b) \u003d a b: GCD (a, b) .

Example 1

It is necessary to find the LCM of the numbers 126 and 70.

Solution

Let's take a = 126 , b = 70 . Substitute the values in the formula for calculating the least common multiple through the greatest common divisor LCM (a, b) = a · b: GCD (a, b) .

Finds the GCD of the numbers 70 and 126. For this we need the Euclid algorithm: 126 = 70 1 + 56 , 70 = 56 1 + 14 , 56 = 14 4 , hence gcd (126 , 70) = 14 .

Let's calculate the LCM: LCM (126, 70) = 126 70: GCD (126, 70) = 126 70: 14 = 630.

Answer: LCM (126, 70) = 630.

Example 2

Find the nok of the numbers 68 and 34.

Solution

GCD in this case is easy to find, since 68 is divisible by 34. Calculate the least common multiple using the formula: LCM (68, 34) = 68 34: GCD (68, 34) = 68 34: 34 = 68.

Answer: LCM(68, 34) = 68.

In this example, we used the rule for finding the least common multiple of positive integers a and b: if the first number is divisible by the second, then the LCM of these numbers will be equal to the first number.

Finding the LCM by Factoring Numbers into Prime Factors

Now let's look at a way to find the LCM, which is based on the decomposition of numbers into prime factors.

Definition 2

To find the least common multiple, we need to perform a number of simple steps:

we make up the product of all prime factors of numbers for which we need to find the LCM;
we exclude all prime factors from their obtained products;
the product obtained after eliminating the common prime factors will be equal to the LCM of the given numbers.

This way of finding the least common multiple is based on the equality LCM (a , b) = a · b: GCM (a , b) . If you look at the formula, it will become clear: the product of the numbers a and b is equal to the product of all factors that are involved in the expansion of these two numbers. In this case, the GCD of two numbers is equal to the product of all prime factors that are simultaneously present in the factorizations of these two numbers.

Example 3

We have two numbers 75 and 210 . We can factor them out as follows: 75 = 3 5 5 And 210 = 2 3 5 7. If you make the product of all the factors of the two original numbers, you get: 2 3 3 5 5 5 7.

If we exclude the factors common to both numbers 3 and 5, we get a product of the following form: 2 3 5 5 7 = 1050. This product will be our LCM for the numbers 75 and 210.

Example 4

Find the LCM of numbers 441 And 700 , decomposing both numbers into prime factors.

Solution

Let's find all the prime factors of the numbers given in the condition:

441 147 49 7 1 3 3 7 7

700 350 175 35 7 1 2 2 5 5 7

We get two chains of numbers: 441 = 3 3 7 7 and 700 = 2 2 5 5 7 .

The product of all the factors that participated in the expansion of these numbers will look like: 2 2 3 3 5 5 7 7 7. Let's find the common factors. This number is 7 . We exclude it from the general product: 2 2 3 3 5 5 7 7. It turns out that NOC (441 , 700) = 2 2 3 3 5 5 7 7 = 44 100.

Answer: LCM (441 , 700) = 44 100 .

Let us give one more formulation of the method for finding the LCM by decomposing numbers into prime factors.

Definition 3

Previously, we excluded from the total number of factors common to both numbers. Now we will do it differently:

Let's decompose both numbers into prime factors:
add to the product of the prime factors of the first number the missing factors of the second number;
we get the product, which will be the desired LCM of two numbers.

Example 5

Let's go back to the numbers 75 and 210 , for which we already looked for the LCM in one of the previous examples. Let's break them down into simple factors: 75 = 3 5 5 And 210 = 2 3 5 7. To the product of factors 3 , 5 and 5 number 75 add the missing factors 2 And 7 numbers 210 . We get: 2 3 5 5 7 . This is the LCM of the numbers 75 and 210.

Example 6

It is necessary to calculate the LCM of the numbers 84 and 648.

Solution

Let's decompose the numbers from the condition into prime factors: 84 = 2 2 3 7 And 648 = 2 2 2 3 3 3 3. Add to the product of the factors 2 , 2 , 3 and 7 numbers 84 missing factors 2 , 3 , 3 and
3 numbers 648 . We get the product 2 2 2 3 3 3 3 7 = 4536 . This is the least common multiple of 84 and 648.

Answer: LCM (84, 648) = 4536.

Finding the LCM of three or more numbers

Regardless of how many numbers we are dealing with, the algorithm of our actions will always be the same: we will consistently find the LCM of two numbers. There is a theorem for this case.

Theorem 1

Suppose we have integers a 1 , a 2 , … , a k. NOC m k of these numbers is found in sequential calculation m 2 = LCM (a 1 , a 2) , m 3 = LCM (m 2 , a 3) , … , m k = LCM (m k − 1 , a k) .

Now let's look at how the theorem can be applied to specific problems.

Example 7

You need to calculate the least common multiple of the four numbers 140 , 9 , 54 and 250 .

Solution

Let's introduce the notation: a 1 \u003d 140, a 2 \u003d 9, a 3 \u003d 54, a 4 \u003d 250.

Let's start by calculating m 2 = LCM (a 1 , a 2) = LCM (140 , 9) . Let's use the Euclidean algorithm to calculate the GCD of the numbers 140 and 9: 140 = 9 15 + 5 , 9 = 5 1 + 4 , 5 = 4 1 + 1 , 4 = 1 4 . We get: GCD(140, 9) = 1, LCM(140, 9) = 140 9: GCD(140, 9) = 140 9: 1 = 1260. Therefore, m 2 = 1 260 .

Now let's calculate according to the same algorithm m 3 = LCM (m 2 , a 3) = LCM (1 260 , 54) . In the course of calculations, we get m 3 = 3 780.

It remains for us to calculate m 4 \u003d LCM (m 3, a 4) \u003d LCM (3 780, 250) . We act according to the same algorithm. We get m 4 \u003d 94 500.

The LCM of the four numbers from the example condition is 94500 .

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

As you can see, the calculations are simple, but quite laborious. To save time, you can go the other way.

Definition 4

We offer you the following algorithm of actions:

decompose all numbers into prime factors;
to the product of the factors of the first number, add the missing factors from the product of the second number;
add the missing factors of the third number to the product obtained at the previous stage, etc.;
the resulting product will be the least common multiple of all numbers from the condition.

Example 8

It is necessary to find the LCM of five numbers 84 , 6 , 48 , 7 , 143 .

Solution

Let's decompose all five numbers into prime factors: 84 = 2 2 3 7 , 6 = 2 3 , 48 = 2 2 2 2 3 , 7 , 143 = 11 13 . Prime numbers, which is the number 7, cannot be factored into prime factors. Such numbers coincide with their decomposition into prime factors.

Now let's take the product of the prime factors 2, 2, 3 and 7 of the number 84 and add to them the missing factors of the second number. We have decomposed the number 6 into 2 and 3. These factors are already in the product of the first number. Therefore, we omit them.

We continue to add the missing multipliers. We turn to the number 48, from the product of prime factors of which we take 2 and 2. Then we add a simple factor of 7 from the fourth number and factors of 11 and 13 of the fifth. We get: 2 2 2 2 3 7 11 13 = 48,048. This is the least common multiple of the five original numbers.

Answer: LCM (84, 6, 48, 7, 143) = 48,048.

Finding the Least Common Multiple of Negative Numbers

In order to find the least common multiple of negative numbers, these numbers must first be replaced by numbers with the opposite sign, and then the calculations should be carried out according to the above algorithms.

Example 9

LCM(54, −34) = LCM(54, 34) and LCM(−622,−46, −54, −888) = LCM(622, 46, 54, 888) .

Such actions are permissible due to the fact that if it is accepted that a And − a- opposite numbers
then the set of multiples a coincides with the set of multiples of a number − a.

Example 10

It is necessary to calculate the LCM of negative numbers − 145 And − 45 .

Solution

Let's change the numbers − 145 And − 45 to their opposite numbers 145 And 45 . Now, using the algorithm, we calculate the LCM (145, 45) = 145 45: GCD (145, 45) = 145 45: 5 = 1 305, having previously determined the GCD using the Euclid algorithm.

We get that the LCM of numbers − 145 and − 45 equals 1 305 .

Answer: LCM (− 145 , − 45) = 1 305 .

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Greatest Common Divisor

Definition 2

If a natural number a is divisible by a natural number $b$, then $b$ is called a divisor of $a$, and the number $a$ is called a multiple of $b$.

Let $a$ and $b$ be natural numbers. The number $c$ is called a common divisor for both $a$ and $b$.

The set of common divisors of the numbers $a$ and $b$ is finite, since none of these divisors can be greater than $a$. This means that among these divisors there is the largest one, which is called the greatest common divisor of the numbers $a$ and $b$, and the notation is used to denote it:

$gcd \ (a;b) \ or \ D \ (a;b)$

To find the greatest common divisor of two numbers:

Find the product of the numbers found in step 2. The resulting number will be the desired greatest common divisor.

Example 1

Find the gcd of the numbers $121$ and $132.$

$242=2\cdot 11\cdot 11$

$132=2\cdot 2\cdot 3\cdot 11$

Choose the numbers that are included in the expansion of these numbers

$242=2\cdot 11\cdot 11$

$132=2\cdot 2\cdot 3\cdot 11$

Find the product of the numbers found in step 2. The resulting number will be the desired greatest common divisor.

$gcd=2\cdot 11=22$

Example 2

Find the GCD of monomials $63$ and $81$.

We will find according to the presented algorithm. For this:

Let's decompose numbers into prime factors

$63=3\cdot 3\cdot 7$

$81=3\cdot 3\cdot 3\cdot 3$

We select the numbers that are included in the expansion of these numbers

$63=3\cdot 3\cdot 7$

$81=3\cdot 3\cdot 3\cdot 3$

Let's find the product of the numbers found in step 2. The resulting number will be the desired greatest common divisor.

$gcd=3\cdot 3=9$

You can find the GCD of two numbers in another way, using the set of divisors of numbers.

Example 3

Find the gcd of the numbers $48$ and $60$.

Solution:

Find the set of divisors of $48$: $\left$(\rm 1,2,3.4.6,8,12,16,24,48)\right$$

Now let's find the set of divisors of $60$:$\ \left$(\rm 1,2,3,4,5,6,10,12,15,20,30,60)\right$$

Let's find the intersection of these sets: $\left$(\rm 1,2,3,4,6,12)\right$$ - this set will determine the set of common divisors of the numbers $48$ and $60$. The largest element in this set will be the number $12$. So the greatest common divisor of $48$ and $60$ is $12$.

Definition of NOC

Definition 3

common multiple of natural numbers$a$ and $b$ is a natural number that is a multiple of both $a$ and $b$.

Common multiples of numbers are numbers that are divisible by the original without a remainder. For example, for the numbers $25$ and $50$, the common multiples will be the numbers $50,100,150,200$, etc.

The least common multiple will be called the least common multiple and denoted by LCM$(a;b)$ or K$(a;b).$

To find the LCM of two numbers, you need:

Decompose numbers into prime factors
Write out the factors that are part of the first number and add to them the factors that are part of the second and do not go to the first

Example 4

Find the LCM of the numbers $99$ and $77$.

We will find according to the presented algorithm. For this

Decompose numbers into prime factors

$99=3\cdot 3\cdot 11$

Write down the factors included in the first

add to them factors that are part of the second and do not go to the first

Find the product of the numbers found in step 2. The resulting number will be the desired least common multiple

$LCC=3\cdot 3\cdot 11\cdot 7=693$

Compiling lists of divisors of numbers is often very time consuming. There is a way to find GCD called Euclid's algorithm.

Statements on which Euclid's algorithm is based:

If $a$ and $b$ are natural numbers, and $a\vdots b$, then $D(a;b)=b$

If $a$ and $b$ are natural numbers such that $b

Using $D(a;b)= D(a-b;b)$, we can successively decrease the numbers under consideration until we reach a pair of numbers such that one of them is divisible by the other. Then the smaller of these numbers will be the desired greatest common divisor for the numbers $a$ and $b$.

Properties of GCD and LCM

Any common multiple of $a$ and $b$ is divisible by K$(a;b)$
If $a\vdots b$ , then K$(a;b)=a$

If K$(a;b)=k$ and $m$-natural number, then K$(am;bm)=km$

If $d$ is a common divisor for $a$ and $b$, then K($\frac(a)(d);\frac(b)(d)$)=$\ \frac(k)(d) $

If $a\vdots c$ and $b\vdots c$ , then $\frac(ab)(c)$ is a common multiple of $a$ and $b$

For any natural numbers $a$ and $b$ the equality

$D(a;b)\cdot K(a;b)=ab$

Any common divisor of $a$ and $b$ is a divisor of $D(a;b)$

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

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Enter numbers in the input field
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press the button "Find GCD and NOC"

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Numbers are entered separated by spaces, dots or commas
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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) :

We factorize both numbers: 28 = 1 2 2 7 , 36 = 1 2 2 3 3
We find common factors, that is, those that both numbers have: 1, 2 and 2.
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:

Find the product of the numbers 28 and 36: 28 36 = 1008
gcd(28, 36) is already known to be 4
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 relationship: 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.

First, let's factorize the numbers: 12 = 1 2 2 3 , 32 = 1 2 2 2 2 2 , 36 = 1 2 2 3 3 .
Let's find common factors: 1, 2 and 2 .
Their product will give gcd: 1 2 2 = 4
Now let's find the LCM: for this we first find the LCM(12, 32): 12 32 / 4 = 96 .
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 .
LCM(12, 32, 36) = 96 36 / 12 = 288 .

To understand how to calculate the LCM, you should first determine the meaning of the term "multiple".

A multiple of A is a natural number that is divisible by A without remainder. Thus, 15, 20, 25, and so on can be considered multiples of 5.

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 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 evenly divisible by all these numbers.

To find the NOC, you can use several methods.

For small numbers, it is convenient to write out in a line all the multiples of these numbers until a common one is found among them. Multiples are denoted in the record with a 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, ...)

So, you can see that the least common multiple of the numbers 4 and 6 is the number 24. This entry is performed 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 way to calculate the LCM.

To complete the task, it is necessary to decompose the proposed numbers into prime factors.

First you need to write out the expansion of the largest of the numbers in a line, and below it - the rest.

In the expansion of each number, there may be a different number of factors.

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

In the expansion of the smaller number, one should underline the factors that are missing in the expansion of the first largest number, and then add them to it. In the presented example, a deuce is missing.

Now we 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, which are not included in the decomposition of the larger number, will be the least common multiple.

To find the LCM of three or more numbers, all of them should be decomposed 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 deuces from the decomposition of sixteen were not included in the factorization of a larger number (one is in the decomposition of twenty-four).

Thus, they need to be added to the decomposition 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, NOCs of twelve and twenty-four would be twenty-four.

If it is necessary to find the least common multiple of coprime numbers that do not have the same divisors, then their LCM will be equal to their product.

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

Second number: b=

Digit separator No space separator " ´

Result:

Greatest Common Divisor gcd( a,b)=6

Least common multiple of LCM( a,b)=468

The largest natural number by which the numbers a and b are divisible without remainder is called greatest common divisor(gcd) of these numbers. Denoted gcd(a,b), (a,b), gcd(a,b) or hcf(a,b).

Least common multiple(LCM) of two integers a and b is the smallest natural number that is divisible by a and b without a remainder. Denoted LCM(a,b), or lcm(a,b).

Integers a and b are called coprime if they have no common divisors other than +1 and −1.

Greatest Common Divisor

Let two positive numbers be given a 1 and a 2 1). It is required to find a common divisor of these numbers, i.e. find such a number λ , which divides the numbers a 1 and a 2 at the same time. Let's describe the algorithm.

1) In this article, the word number will mean an integer.

Let a 1 ≥ a 2 and let

Where m 1 , a 3 are some integers, a 3 <a 2 (remainder from division a 1 on a 2 should be less a 2).

Let's pretend that λ divides a 1 and a 2 , then λ divides m 1 a 2 and λ divides a 1 −m 1 a 2 =a 3 (Assertion 2 of the article "Divisibility of numbers. Sign of divisibility"). It follows that every common divisor a 1 and a 2 is a common divisor a 2 and a 3 . The converse is also true if λ common divisor a 2 and a 3 , then m 1 a 2 and a 1 =m 1 a 2 +a 3 are also divided into λ . Hence the common divisor a 2 and a 3 is also a common divisor a 1 and a 2. Because a 3 <a 2 ≤a 1 , then we can say that the solution to the problem of finding a common divisor of numbers a 1 and a 2 reduced to a simpler problem of finding a common divisor of numbers a 2 and a 3 .

If a 3 ≠0, then we can divide a 2 on a 3 . Then

Where m 1 and a 4 are some integers, ( a 4 remainder of division a 2 on a 3 (a 4 <a 3)). By similar reasoning, we come to the conclusion that the common divisors of numbers a 3 and a 4 is the same as common divisors of numbers a 2 and a 3 , and also with common divisors a 1 and a 2. Because a 1 , a 2 , a 3 , a 4 , ... numbers that are constantly decreasing, and since there is a finite number of integers between a 2 and 0, then at some step n, remainder of the division a n on a n+1 will be equal to zero ( a n+2=0).

Every common divisor λ numbers a 1 and a 2 is also a divisor of numbers a 2 and a 3 , a 3 and a 4 , .... a n and a n+1 . The converse is also true, common divisors of numbers a n and a n+1 are also divisors of numbers a n−1 and a n , .... , a 2 and a 3 , a 1 and a 2. But the common divisor a n and a n+1 is a number a n+1 , because a n and a n+1 are divisible by a n+1 (recall that a n+2=0). Hence a n+1 is also a divisor of numbers a 1 and a 2 .

Note that the number a n+1 is the greatest number divisor a n and a n+1 , since the greatest divisor a n+1 is itself a n+1 . If a n + 1 can be represented as a product of integers, then these numbers are also common divisors of numbers a 1 and a 2. Number a n+1 are called greatest common divisor numbers a 1 and a 2 .

Numbers a 1 and a 2 can be both positive and negative numbers. If one of the numbers is equal to zero, then the greatest common divisor of these numbers will be equal to the absolute value of the other number. The greatest common divisor of zero numbers is not defined.

The above algorithm is called Euclid's algorithm to find the greatest common divisor of two integers.

An example of finding the greatest common divisor of two numbers

Find the greatest common divisor of two numbers 630 and 434.

Step 1. Divide the number 630 by 434. The remainder is 196.
Step 2. Divide the number 434 by 196. The remainder is 42.
Step 3. Divide the number 196 by 42. The remainder is 28.
Step 4. Divide the number 42 by 28. The remainder is 14.
Step 5. Divide the number 28 by 14. The remainder is 0.

At step 5, the remainder of the division is 0. Therefore, the greatest common divisor of the numbers 630 and 434 is 14. Note that the numbers 2 and 7 are also divisors of the numbers 630 and 434.

Coprime numbers

Definition 1. Let the greatest common divisor of numbers a 1 and a 2 is equal to one. Then these numbers are called coprime numbers that do not have a common divisor.

Theorem 1. If a 1 and a 2 relatively prime numbers, and λ some number, then any common divisor of numbers λa 1 and a 2 is also a common divisor of numbers λ And a 2 .

Proof. Consider Euclid's algorithm for finding the greatest common divisor of numbers a 1 and a 2 (see above).

It follows from the conditions of the theorem that the greatest common divisor of numbers a 1 and a 2 , and therefore a n and a n+1 is 1. I.e. a n+1=1.

Let's multiply all these equalities by λ , Then

Let the common divisor a 1 λ And a 2 is δ . Then δ enters as a factor in a 1 λ , m 1 a 2 λ and in a 1 λ -m 1 a 2 λ =a 3 λ (See "Divisibility of numbers", Statement 2). Further δ enters as a factor in a 2 λ And m 2 a 3 λ , and hence enters as a factor in a 2 λ -m 2 a 3 λ =a 4 λ .

By reasoning in this way, we are convinced that δ enters as a factor in a n−1 λ And m n−1 a n λ , and therefore in a n−1 λ −m n−1 a n λ =a n+1 λ . Because a n+1 =1, then δ enters as a factor in λ . Hence the number δ is a common divisor of numbers λ And a 2 .

Consider special cases of Theorem 1.

Consequence 1. Let a And c prime numbers are relatively b. Then their product ac is a prime number with respect to b.

Really. From Theorem 1 ac And b have the same common divisors as c And b. But the numbers c And b coprime, i.e. have a single common divisor 1. Then ac And b also have a single common divisor 1. Hence ac And b mutually simple.

Consequence 2. Let a And b coprime numbers and let b divides ak. Then b divides and k.

Really. From the assertion condition ak And b have a common divisor b. By virtue of Theorem 1, b must be a common divisor b And k. Hence b divides k.

Corollary 1 can be generalized.

Consequence 3. 1. Let the numbers a 1 , a 2 , a 3 , ..., a m are prime relative to the number b. Then a 1 a 2 , a 1 a 2 · a 3 , ..., a 1 a 2 a 3 ··· a m , the product of these numbers is prime with respect to the number b.

2. Let we have two rows of numbers

such that every number in the first row is prime with respect to every number in the second row. Then the product

It is required to find such numbers that are divisible by each of these numbers.

If the number is divisible by a 1 , then it looks like sa 1 , where s some number. If q is the greatest common divisor of numbers a 1 and a 2 , then

Where s 1 is some integer. Then

is least common multiple of numbers a 1 and a 2 .

a 1 and a 2 coprime, then the least common multiple of the numbers a 1 and a 2:

Find the least common multiple of these numbers.

It follows from the above that any multiple of the numbers a 1 , a 2 , a 3 must be a multiple of numbers ε And a 3 and vice versa. Let the least common multiple of the numbers ε And a 3 is ε 1 . Further, a multiple of numbers a 1 , a 2 , a 3 , a 4 must be a multiple of numbers ε 1 and a 4 . Let the least common multiple of the numbers ε 1 and a 4 is ε 2. Thus, we found out that all multiples of numbers a 1 , a 2 , a 3 ,...,a m coincide with multiples of some specific number ε n , which is called the least common multiple of the given numbers.

In the particular case when the numbers a 1 , a 2 , a 3 ,...,a m coprime, then the least common multiple of the numbers a 1 , a 2 as shown above has the form (3). Further, since a 3 prime with respect to numbers a 1 , a 2 , then a 3 is a prime relative number a 1 · a 2 (Corollary 1). So the least common multiple of the numbers a 1 ,a 2 ,a 3 is a number a 1 · a 2 · a 3 . Arguing in a similar way, we arrive at the following assertions.

Statement 1. Least common multiple of coprime numbers a 1 , a 2 , a 3 ,...,a m is equal to their product a 1 · a 2 · a 3 ··· a m .

Statement 2. Any number that is divisible by each of the coprime numbers a 1 , a 2 , a 3 ,...,a m is also divisible by their product a 1 · a 2 · a 3 ··· a m .