Definition. Greatest natural number, by which the numbers a and b are divided without remainder, is called greatest common divisor (GCD) these numbers.
Let's find the largest common divisor numbers 24 and 35.
The divisors of 24 are the numbers 1, 2, 3, 4, 6, 8, 12, 24, and the divisors of 35 are the numbers 1, 5, 7, 35.
We see that the numbers 24 and 35 have only one common divisor - the number 1. Such numbers are called mutually prime.
Definition. Natural numbers are called mutually prime, if their greatest common divisor (GCD) is 1.
Greatest Common Divisor (GCD) can be found without writing out all the divisors of the given numbers.
Let's factor the numbers 48 and 36 and get:
48 = 2 * 2 * 2 * 2 * 3, 36 = 2 * 2 * 3 * 3.
From the factors included in the expansion of the first of these numbers, we cross out those that are not included in the expansion of the second number (i.e., two twos).
The factors remaining are 2 * 2 * 3. Their product is equal to 12. This number is the greatest common divisor of the numbers 48 and 36. The greatest common divisor of three or more numbers is also found.
To find greatest common divisor
2) from the factors included in the expansion of one of these numbers, cross out those that are not included in the expansion of other numbers;
3) find the product of the remaining factors.
If all given numbers are divisible by one of them, then this number is greatest common divisor given numbers.
For example, the greatest common divisor of the numbers 15, 45, 75 and 180 is the number 15, since all other numbers are divisible by it: 45, 75 and 180.
Least common multiple (LCM)
Definition. Least common multiple (LCM) natural numbers a and b is the smallest natural number that is a multiple of both a and b. The least common multiple (LCM) of the numbers 75 and 60 can be found without writing down the multiples of these numbers in a row. To do this, let's decompose 75 and 60 into prime factors: 75 = 3 * 5 * 5, and 60 = 2 * 2 * 3 * 5.
Let's write out the factors included in the expansion of the first of these numbers, and add to them the missing factors 2 and 2 from the expansion of the second number (i.e., we combine the factors).
We get five factors 2 * 2 * 3 * 5 * 5, the product of which is 300. This number is the least common multiple of the numbers 75 and 60.
They also find the least common multiple of three or more numbers.
To find least common multiple several natural numbers, you need:
1) factor them into prime factors;
2) write down the factors included in the expansion of one of the numbers;
3) add to them the missing factors from the expansions of the remaining numbers;
4) find the product of the resulting factors.
Note that if one of these numbers is divisible by all other numbers, then this number is the least common multiple of these numbers.
For example, the least common multiple of the numbers 12, 15, 20, and 60 is 60 because it is divisible by all of those numbers.
Pythagoras (VI century BC) and his students studied the question of the divisibility of numbers. Number, equal to the sum They called all its divisors (without the number itself) a perfect number. For example, the numbers 6 (6 = 1 + 2 + 3), 28 (28 = 1 + 2 + 4 + 7 + 14) are perfect. The next perfect numbers are 496, 8128, 33,550,336. The Pythagoreans only knew the first three perfect numbers. The fourth - 8128 - became known in the 1st century. n. e. The fifth - 33,550,336 - was found in the 15th century. By 1983, 27 perfect numbers were already known. But scientists still don’t know whether there are odd perfect numbers or whether there is a largest perfect number.
The interest of ancient mathematicians in prime numbers stems from the fact that any number is either prime or can be represented as a product prime numbers, i.e. prime numbers are like bricks from which the rest of the natural numbers are built.
You probably noticed that prime numbers in the series of natural numbers occur unevenly - in some parts of the series there are more of them, in others - less. But the further we move along the number series, the less common prime numbers are. The question arises: is there a last (largest) prime number? The ancient Greek mathematician Euclid (3rd century BC), in his book “Elements”, which was the main textbook of mathematics for two thousand years, proved that there are infinitely many prime numbers, i.e. behind each prime number there is an even greater prime number.
To find prime numbers, another Greek mathematician of the same time, Eratosthenes, came up with this method. He wrote down all the numbers from 1 to some number, and then crossed out one, which is neither a prime nor a composite number, then crossed out through one all the numbers coming after 2 (numbers that are multiples of 2, i.e. 4, 6 , 8, etc.). The first remaining number after 2 was 3. Then, after two, all numbers coming after 3 (numbers that are multiples of 3, i.e. 6, 9, 12, etc.) were crossed out. in the end only the prime numbers remained uncrossed.
The material presented below is a logical continuation of the theory from the article entitled LCM - least common multiple, definition, examples, connection between LCM and GCD. Here we will talk about finding the least common multiple (LCM), And Special attention Let's focus on solving examples. First, we will show how the LCM of two numbers is calculated using the GCD of these numbers. Next, we'll look at finding the least common multiple by factoring numbers into prime factors. After this, we will focus on finding the LCM of three or more numbers, and also pay attention to calculating the LCM of negative numbers.
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Calculating Least Common Multiple (LCM) via GCD
One way to find the least common multiple is based on the relationship between LCM and GCD. Existing connection between LCM and GCD allows you to calculate the least common multiple of two positive integers using a known greatest common divisor. The corresponding formula is LCM(a, b)=a b:GCD(a, b) . Let's consider examples of finding the LCM using the given formula.
Example.
Find the least common multiple of two numbers 126 and 70.
Solution.
In this example a=126 , b=70 . Let us use the connection between LCM and GCD, expressed by the formula LCM(a, b)=a b:GCD(a, b). That is, first we have to find the greatest common divisor of the numbers 70 and 126, after which we can calculate the LCM of these numbers using the written formula.
Let's find GCD(126, 70) using the Euclidean algorithm: 126=70·1+56, 70=56·1+14, 56=14·4, therefore, GCD(126, 70)=14.
Now we find the required least common multiple: GCD(126, 70)=126·70:GCD(126, 70)= 126·70:14=630.
Answer:
LCM(126, 70)=630 .
Example.
What is LCM(68, 34) equal to?
Solution.
Because 68 is divisible by 34, then GCD(68, 34)=34. Now we calculate the least common multiple: GCD(68, 34)=68·34:GCD(68, 34)= 68·34:34=68.
Answer:
LCM(68, 34)=68 .
Note that the previous example fits the following rule for finding the LCM for positive integers a and b: if the number a is divisible by b, then the least common multiple of these numbers is a.
Finding 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 you compose a product from all the prime factors of given numbers, and then exclude from this product all the common prime factors present in the decompositions of the given numbers, then the resulting product will be equal to the least common multiple of the given numbers.
The stated rule for finding the LCM follows from the equality LCM(a, b)=a b:GCD(a, b). Indeed, the product of numbers a and b is equal to the product of all factors involved in the expansion of numbers a and b. In turn, GCD(a, b) is equal to the product of all prime factors simultaneously present in the expansions of numbers a and b (as described in the section on finding GCD using the expansion of numbers into prime factors).
Let's give an example. Let us know that 75=3·5·5 and 210=2·3·5·7. Let's compose the product from all the factors of these expansions: 2·3·3·5·5·5·7 . Now from this product we exclude all the factors present in both the expansion of the number 75 and the expansion of the number 210 (these factors are 3 and 5), then the product will take the form 2·3·5·5·7. The value of this product is equal to the least common multiple of 75 and 210, that is, NOC(75, 210)= 2·3·5·5·7=1,050.
Example.
Factor the numbers 441 and 700 into prime factors and find the least common multiple of these numbers.
Solution.
Let's factor the numbers 441 and 700 into prime factors:
We get 441=3·3·7·7 and 700=2·2·5·5·7.
Now let’s create a product from all the factors involved in the expansion of these numbers: 2·2·3·3·5·5·7·7·7. Let us exclude from this product all factors that are simultaneously present in both expansions (there is only one such factor - this is the number 7): 2·2·3·3·5·5·7·7. Thus, LCM(441, 700)=2·2·3·3·5·5·7·7=44 100.
Answer:
NOC(441, 700)= 44 100 .
The rule for finding the LCM using factorization of numbers into prime factors can be formulated a little differently. If the missing factors from the expansion of number b are added to the factors from the expansion of the number a, then the value of the resulting product will be equal to the least common multiple of the numbers a and b.
For example, let's take the same numbers 75 and 210, their decompositions into prime factors are as follows: 75=3·5·5 and 210=2·3·5·7. To the factors 3, 5 and 5 from the expansion of the number 75 we add the missing factors 2 and 7 from the expansion of the number 210, we obtain the product 2·3·5·5·7, the value of which is equal to LCM(75, 210).
Example.
Find the least common multiple of 84 and 648.
Solution.
We first obtain the decompositions of the numbers 84 and 648 into prime factors. They look like 84=2·2·3·7 and 648=2·2·2·3·3·3·3. To the factors 2, 2, 3 and 7 from the expansion of the number 84 we add the missing factors 2, 3, 3 and 3 from the expansion of the number 648, we obtain the product 2 2 2 3 3 3 3 7, which is equal to 4 536 . Thus, the desired least common multiple of 84 and 648 is 4,536.
Answer:
LCM(84, 648)=4,536 .
Finding the LCM of three or more numbers
The least common multiple of three or more numbers can be found by sequentially finding the LCM of two numbers. Let us recall the corresponding theorem, which gives a way to find the LCM of three or more numbers.
Theorem.
Let positive integer numbers a 1 , a 2 , …, a k be given, the least common multiple m k of these numbers is found by sequentially calculating m 2 = LCM(a 1 , a 2) , m 3 = LCM(m 2 , a 3) , … , m k = LCM(m k−1 , a k) .
Let's consider the application of this theorem using the example of finding the least common multiple of four numbers.
Example.
Find the LCM of four numbers 140, 9, 54 and 250.
Solution.
In this example, a 1 =140, a 2 =9, a 3 =54, a 4 =250.
First we find m 2 = LOC(a 1 , a 2) = LOC(140, 9). To do this, using the Euclidean algorithm, we determine GCD(140, 9), we have 140=9·15+5, 9=5·1+4, 5=4·1+1, 4=1·4, therefore, GCD(140, 9)=1 , from where GCD(140, 9)=140 9:GCD(140, 9)= 140·9:1=1,260. That is, m 2 =1 260.
Now we find m 3 = LOC (m 2 , a 3) = LOC (1 260, 54). Let's calculate it through GCD(1 260, 54), which we also determine using the Euclidean algorithm: 1 260=54·23+18, 54=18·3. Then gcd(1,260, 54)=18, from which gcd(1,260, 54)= 1,260·54:gcd(1,260, 54)= 1,260·54:18=3,780. That is, m 3 =3 780.
All that remains is to find m 4 = LOC(m 3, a 4) = LOC(3 780, 250). To do this, we find GCD(3,780, 250) using the Euclidean algorithm: 3,780=250·15+30, 250=30·8+10, 30=10·3. Therefore, GCM(3,780, 250)=10, whence GCM(3,780, 250)= 3 780 250: GCD(3 780, 250)= 3,780·250:10=94,500. That is, m 4 =94,500.
So the least common multiple of the original four numbers is 94,500.
Answer:
LCM(140, 9, 54, 250)=94,500.
In many cases, it is convenient to find the least common multiple of three or more numbers using prime factorizations of the given numbers. In this case, you should adhere to the following rule. The least common multiple of several numbers is equal to the product, which is composed as follows: the missing factors from the expansion of the second number are added to all factors from the expansion of the first number, the missing factors from the expansion of the third number are added to the resulting factors, and so on.
Let's look at an example of finding the least common multiple using prime factorization.
Example.
Find the least common multiple of the five numbers 84, 6, 48, 7, 143.
Solution.
First, we obtain decompositions of these numbers into prime factors: 84=2·2·3·7, 6=2·3, 48=2·2·2·2·3, 7 (7 is a prime number, it coincides with its decomposition into prime factors) and 143=11·13.
To find the LCM of these numbers, to the factors of the first number 84 (they are 2, 2, 3 and 7), you need to add the missing factors from the expansion of the second number 6. The decomposition of the number 6 does not contain missing factors, since both 2 and 3 are already present in the decomposition of the first number 84. Next, to the factors 2, 2, 3 and 7 we add the missing factors 2 and 2 from the expansion of the third number 48, we get a set of factors 2, 2, 2, 2, 3 and 7. There will be no need to add multipliers to this set in the next step, since 7 is already contained in it. Finally, to the factors 2, 2, 2, 2, 3 and 7 we add the missing factors 11 and 13 from the expansion of the number 143. We get the product 2·2·2·2·3·7·11·13, which is equal to 48,048.
Let's continue the conversation about the least common multiple, which we started in the section “LCM - least common multiple, definition, examples.” In this topic, we will look at ways to find the LCM for three or more numbers, and we will look at the question of how to find the LCM of a negative number.
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Calculating Least Common Multiple (LCM) via GCD
We have already established the relationship between the least common multiple and the greatest common divisor. Now let's learn how to determine the LCM through 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) = a · b: GCD (a, b).
Example 1
You need to find the LCM of the numbers 126 and 70.
Solution
Let's take a = 126, b = 70. Let's substitute the values into 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 numbers 70 and 126. For this we need the Euclidean algorithm: 126 = 70 1 + 56, 70 = 56 1 + 14, 56 = 14 4, therefore GCD (126 , 70) = 14 .
Let's calculate the LCM: LCD (126, 70) = 126 70: GCD (126, 70) = 126 70: 14 = 630.
Answer: LCM(126, 70) = 630.
Example 2
Find the number 68 and 34.
Solution
GCD in in this case This is not difficult, since 68 is divisible by 34. Let's 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, the LCM of those numbers will be equal to the first number.
Finding the LCM by factoring numbers into prime factors
Now let's look at the method of finding the LCM, which is based on factoring numbers into prime factors.
Definition 2
To find the least common multiple, we need to perform a number of simple steps:
- we compose the product of all prime factors of the numbers for which we need to find the LCM;
- we exclude all prime factors from their resulting products;
- the product obtained after eliminating the common prime factors will be equal to the LCM of the given numbers.
This method of finding the least common multiple is based on the equality LCM (a, b) = a · b: GCD (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 the factors that participate in the decomposition 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 as follows: 75 = 3 5 5 And 210 = 2 3 5 7. If you compose 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 , factoring 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 factors that participated in the decomposition of these numbers will have the form: 2 2 3 3 5 5 7 7 7. Let's find common factors. This is the number 7. Let's exclude him from total 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: LOC(441, 700) = 44,100.
Let us give another 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 factor 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 obtain the product, which will be the desired LCM of two numbers.
Example 5
Let's return 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 numbers 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 factor the numbers from the condition into simple factors: 84 = 2 2 3 7 And 648 = 2 2 2 3 3 3 3. Let's add to the product 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) = 4,536.
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 sequentially find the LCM of two numbers. There is a theorem for this case.
Theorem 1
Let's assume we have integers a 1 , a 2 , … , a k. NOC m k these numbers are found by sequentially calculating m 2 = LCM (a 1, a 2), m 3 = LCM (m 2, a 3), ..., m k = LCM (m k − 1, a k).
Now let's look at how the theorem can be applied to solve specific problems.
Example 7
You need to calculate the least common multiple of four numbers 140, 9, 54 and 250 .
Solution
Let us introduce the notation: a 1 = 140, a 2 = 9, a 3 = 54, a 4 = 250.
Let's start by calculating m 2 = LCM (a 1 , a 2) = LCM (140, 9). Let's apply 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, GCD (140, 9) = 140 9: GCD (140, 9) = 140 9: 1 = 1,260. Therefore, m 2 = 1,260.
Now let’s calculate using the same algorithm m 3 = LCM (m 2 , a 3) = LCM (1 260, 54). During the calculations we obtain m 3 = 3 780.
We just have to calculate m 4 = LCM (m 3 , a 4) = LCM (3 780, 250). We follow the same algorithm. We get m 4 = 94 500.
The LCM of the four numbers from the example condition is 94500.
Answer: NOC (140, 9, 54, 250) = 94,500.
As you can see, the calculations are simple, but quite labor-intensive. To save time, you can go another way.
Definition 4
We offer you the following algorithm of actions:
- we decompose all numbers into prime factors;
- to the product of the factors of the first number we add the missing factors from the product of the second number;
- to the product obtained at the previous stage we add the missing factors of the third number, etc.;
- the resulting product will be the least common multiple of all numbers from the condition.
Example 8
You need to find the LCM of five numbers 84, 6, 48, 7, 143.
Solution
Let's factor 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 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. Let's move on to the number 48, from the product of whose prime factors we take 2 and 2. Then we add the prime factor of 7 from the fourth number and the 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 original five 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 must be carried out using 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 we accept that a And − a– opposite numbers,
then the set of multiples of a number a matches 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 replace 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 Euclidean algorithm.
We get that the LCM of the numbers is − 145 and − 45 equals 1 305 .
Answer: LCM (− 145, − 45) = 1,305.
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To understand how to calculate the LCM, you must first determine the meaning of the term “multiple.”
A multiple of A is a natural number that is divisible by A without a remainder. Thus, numbers that are multiples of 5 can be considered 15, 20, 25, and so on.
There can be divisors of a specific number limited quantity, but there are an infinite number of multiples.
A common multiple of natural numbers is a number that is divisible by them without leaving a remainder.
How to find the least common multiple of numbers
The least common multiple (LCM) of numbers (two, three or more) is the smallest natural number that is divisible by all these numbers.
To find the LOC, you can use several methods.
For small numbers, it is convenient to write down all the multiples of these numbers on a line until you find something common among them. Multiples are denoted by the capital letter K.
For example, multiples of 4 can be written like this:
K (4) = (8,12, 16, 20, 24, ...)
K (6) = (12, 18, 24, ...)
Thus, you can see that the least common multiple of the numbers 4 and 6 is the number 24. This notation is done as follows:
LCM(4, 6) = 24
If the numbers are large, find the common multiple of three or more numbers, then it is better to use another method of calculating the LCM.
To complete the task, you need to factor the given numbers into prime factors.
First you need to write down the decomposition of the largest number on a line, and below it - the rest.
The decomposition of each number may contain a different number of factors.
For example, let's factor the numbers 50 and 20 into prime factors.
In the expansion of the smaller number, it is necessary to emphasize the factors that are absent in the expansion of the first one. large number, and then add them to it. In the example presented, a two is missing.
Now you can calculate the least common multiple of 20 and 50.
LCM(20, 50) = 2 * 5 * 5 * 2 = 100
So, the product of prime factors more and the factors of the second number that were not included in the expansion of the larger number will be the least common multiple.
To find the LCM of three or more numbers, you should factor them all into prime factors, as in the previous case.
As an example, you can find the least common multiple of the numbers 16, 24, 36.
36 = 2 * 2 * 3 * 3
24 = 2 * 2 * 2 * 3
16 = 2 * 2 * 2 * 2
Thus, only two twos from the expansion of sixteen were not included in the factorization of a larger number (one is in the expansion of twenty-four).
Thus, they need to be added to the expansion of a larger number.
LCM(12, 16, 36) = 2 * 2 * 3 * 3 * 2 * 2 = 9
There are special cases of determining the least common multiple. So, if one of the numbers can be divided without a remainder by another, then the larger of these numbers will be the least common multiple.
For example, the LCM of twelve and twenty-four is twenty-four.
If it is necessary to find the least common multiple of coprime numbers that do not have identical divisors, then their LCM will be equal to their product.
For example, LCM (10, 11) = 110.
Second number: b=
Thousand separator Without space separator "´
Result:
Greatest common divisor gcd( a,b)=6
Least common multiple of LCM( a,b)=468
The largest natural number by which numbers a and b are divided without remainder is called greatest common divisor(GCD) of these numbers. Denoted by gcd(a,b), (a,b), gcd(a,b) or hcf(a,b).
Least common multiple The 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).
The integers a and b are called mutually prime, 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 the common divisor of these numbers, i.e. find such a number λ , which divides numbers a 1 and a 2 at the same time. Let's describe the algorithm.
1) In this article, the word number will be understood as an integer.
Let a 1 ≥ a 2 and let
Where m 1 , a 3 are some integers, a 3 <a 2 (remainder of division a 1 per 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 (Statement 2 of the article “Divisibility of numbers. Divisibility test”). It follows that every common divisor a 1 and a 2 is the common divisor a 2 and a 3. The reverse 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 is also divisible by λ . Therefore 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 the common divisor of numbers a 1 and a 2 reduced to the simpler problem of finding the common divisor of numbers a 2 and a 3 .
If a 3 ≠0, then we can divide a 2 per a 3. Then
,
Where m 1 and a 4 are some integers, ( a 4 remainder from division a 2 per a 3 (a 4 <a 3)). By similar reasoning we come to the conclusion that common divisors of numbers a 3 and a 4 coincides with 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, ... are 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 of numbers 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 (remember 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 largest divisor of numbers 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 is called greatest common divisor numbers a 1 and a 2 .
Numbers a 1 and a 2 can be either positive or 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 undefined.
The above algorithm is called Euclidean 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.
In 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 the numbers a 1 and a 2 is equal to one. Then these numbers are called coprime numbers, having no common divisor.
Theorem 1. If a 1 and a 2 coprime 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 the Euclidean algorithm for finding the greatest common divisor of numbers a 1 and a 2 (see above).
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From the conditions of the theorem it follows that the greatest common divisor of the numbers a 1 and a 2 and therefore a n and a n+1 is 1. That is a n+1 =1.
Let's multiply all these equalities by λ , Then
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Let the common divisor a 1 λ And a 2 yes δ . Then δ is included as a multiplier 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 is included as a multiplier in a 2 λ And m 2 a 3 λ , and, therefore, is a factor in a 2 λ -m 2 a 3 λ =a 4 λ .
Reasoning this way, we are convinced that δ is included as a multiplier 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 is included as a multiplier in λ . Therefore the number δ is the common divisor of numbers λ And a 2 .
Let us 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 relatively simple, i.e. have a single common divisor 1. Then ac And b also have a single common divisor 1. Therefore 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 approval 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 us have two rows of numbers
such that every number in the first series is prime in the ratio of every number in the second series. Then the product
You need to find numbers that are divisible by each of these numbers.
If a number is divisible by a 1, then it has the form 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 multiples of numbers a 1 and a 2 .
a 1 and a 2 are relatively prime, then the least common multiple of the numbers a 1 and a 2:
We need to find the least common multiple of these numbers.
From the above it follows that any multiple of numbers a 1 , a 2 , a 3 must be a multiple of numbers ε And a 3 and back. Let the least common multiple of the numbers ε And a 3 yes ε 1 . Next, multiples 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 yes ε 2. Thus, we found out that all multiples of numbers a 1 , a 2 , a 3 ,...,a m coincide with multiples of a certain number ε n, which is called the least common multiple of the given numbers.
In the special case when the numbers a 1 , a 2 , a 3 ,...,a m are relatively prime, then the least common multiple of the numbers a 1 , a 2, as shown above, has the form (3). Next, since a 3 prime in relation to numbers a 1 , a 2 then a 3 prime number a 1 · a 2 (Corollary 1). Means the least common multiple of numbers a 1 ,a 2 ,a 3 is a number a 1 · a 2 · a 3. Reasoning in a similar way, we arrive at the following statements.
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.