But many natural numbers are evenly divisible by other natural numbers.
For example:
The number 12 is divided by 1, by 2, by 3, by 4, by 6, by 12;
The number 36 is divisible by 1, 2, 3, 4, 6, 12, 18, 36.
The numbers by which the number is evenly divisible (for 12 it is 1, 2, 3, 4, 6 and 12) are called divisors... Natural number divisor a is a natural number that divides a given number a without a remainder. A natural number that has more than two divisors is called composite .
Note that the numbers 12 and 36 have common factors. These are numbers: 1, 2, 3, 4, 6, 12. The largest divisor of these numbers is 12. Common divisor of two given numbers a and b- this is the number by which both given numbers are divisible without a remainder a and b.
Common multiple multiple numbers is a number that is divisible by each of these numbers. For example, the numbers 9, 18 and 45 have a common multiple of 180. But 90 and 360 are also their common multiples. Among all j total multiples, there is always the smallest, in this case it is 90. This number is called the smallestcommon multiple (LCM).
The LCM is always a natural number, which must be greater than the largest of the numbers for which it is determined.
Least Common Multiple (LCM). Properties.
Commutability:
Associativity:
In particular, if and are coprime numbers, then:
Least common multiple of two integers m and n is the divisor of all other common multiples m and n... Moreover, the set of common multiples m, n coincides with the set of multiples for LCM ( m, n).
The asymptotics for can be expressed in terms of some number-theoretic functions.
So, Chebyshev function... And:
This follows from the definition and properties of the Landau function g (n).
What follows from the law of distribution of prime numbers.
Finding the least common multiple (LCM).
LCM ( a, b) can be calculated in several ways:
1. If the greatest common divisor is known, you can use its relationship with the LCM:
2. Let the canonical decomposition of both numbers into prime factors be known:
where p 1, ..., p k- various primes, and d 1, ..., d k and e 1, ..., e k- non-negative integers (they can be zeros if the corresponding prime is absent in the decomposition).
Then LCM ( a,b) is calculated by the formula:
In other words, the LCM decomposition contains all prime factors included in at least one of the number expansions a, b, and the largest of the two exponents of this factor is taken.
Example:
The calculation of the least common multiple of several numbers can be reduced to several consecutive calculations of the LCM of two numbers:
Rule. To find the LCM of a series of numbers, you need:
- to decompose numbers into prime factors;
- transfer the largest expansion into the factors of the desired product (the product of the factors of the largest number of the given ones), and then add the factors from the expansion of other numbers that do not occur in the first number or are in it fewer times;
- the resulting product of prime factors will be the LCM of the given numbers.
Any two or more natural numbers have their own LCM. If the numbers are not multiples of each other or do not have the same factors in the expansion, then their LCM is equal to the product of these numbers.
The prime factors of the number 28 (2, 2, 7) were supplemented with a factor of 3 (number 21), the resulting product (84) will be the smallest number that is divisible by 21 and 28.
The prime factors of the largest number 30 were supplemented with a factor of 5 of the number 25, the resulting product 150 is greater than the largest number 30 and is divided by all the given numbers without a remainder. This is the smallest possible product (150, 250, 300 ...), which is a multiple of all given numbers.
The numbers 2,3,11,37 are simple, so their LCM is equal to the product of the given numbers.
The rule... To calculate the LCM of prime numbers, you need to multiply all these numbers among themselves.
Another option:
To find the least common multiple (LCM) of several numbers, you need:
1) represent each number as the product of its prime factors, for example:
504 = 2 2 2 3 3 7,
2) write down the powers of all prime factors:
504 = 2 2 2 3 3 7 = 2 3 3 2 7 1,
3) write down all the prime divisors (factors) of each of these numbers;
4) choose the highest degree of each of them, found in all expansions of these numbers;
5) multiply these degrees.
Example... Find the LCM of numbers: 168, 180 and 3024.
Solution... 168 = 2 2 2 3 7 = 2 3 3 1 7 1,
180 = 2 2 3 3 5 = 2 2 3 2 5 1,
3024 = 2 2 2 2 3 3 3 7 = 2 4 3 3 7 1.
We write out the greatest powers of all prime factors and multiply them:
LCM = 2 4 3 3 5 1 7 1 = 15 120.
But many natural numbers are evenly divisible by other natural numbers.
For example:
The number 12 is divided by 1, by 2, by 3, by 4, by 6, by 12;
The number 36 is divisible by 1, 2, 3, 4, 6, 12, 18, 36.
The numbers by which the number is evenly divisible (for 12 it is 1, 2, 3, 4, 6 and 12) are called divisors... Natural number divisor a is a natural number that divides a given number a without a remainder. A natural number that has more than two divisors is called composite .
Note that the numbers 12 and 36 have common factors. These are numbers: 1, 2, 3, 4, 6, 12. The largest divisor of these numbers is 12. Common divisor of two given numbers a and b- this is the number by which both given numbers are divisible without a remainder a and b.
Common multiple multiple numbers is a number that is divisible by each of these numbers. For example, the numbers 9, 18 and 45 have a common multiple of 180. But 90 and 360 are also their common multiples. Among all j total multiples, there is always the smallest, in this case it is 90. This number is called the smallestcommon multiple (LCM).
The LCM is always a natural number, which must be greater than the largest of the numbers for which it is determined.
Least Common Multiple (LCM). Properties.
Commutability:
Associativity:
In particular, if and are coprime numbers, then:
Least common multiple of two integers m and n is the divisor of all other common multiples m and n... Moreover, the set of common multiples m, n coincides with the set of multiples for LCM ( m, n).
The asymptotics for can be expressed in terms of some number-theoretic functions.
So, Chebyshev function... And:
This follows from the definition and properties of the Landau function g (n).
What follows from the law of distribution of prime numbers.
Finding the least common multiple (LCM).
LCM ( a, b) can be calculated in several ways:
1. If the greatest common divisor is known, you can use its relationship with the LCM:
2. Let the canonical decomposition of both numbers into prime factors be known:
where p 1, ..., p k- various primes, and d 1, ..., d k and e 1, ..., e k- non-negative integers (they can be zeros if the corresponding prime is absent in the decomposition).
Then LCM ( a,b) is calculated by the formula:
In other words, the LCM decomposition contains all prime factors included in at least one of the number expansions a, b, and the largest of the two exponents of this factor is taken.
Example:
The calculation of the least common multiple of several numbers can be reduced to several consecutive calculations of the LCM of two numbers:
Rule. To find the LCM of a series of numbers, you need:
- to decompose numbers into prime factors;
- transfer the largest expansion into the factors of the desired product (the product of the factors of the largest number of the given ones), and then add the factors from the expansion of other numbers that do not occur in the first number or are in it fewer times;
- the resulting product of prime factors will be the LCM of the given numbers.
Any two or more natural numbers have their LCM. If the numbers are not multiples of each other or do not have the same factors in the expansion, then their LCM is equal to the product of these numbers.
The prime factors of the number 28 (2, 2, 7) were supplemented with a factor of 3 (number 21), the resulting product (84) will be the smallest number that is divisible by 21 and 28.
The prime factors of the largest number 30 were supplemented with a factor of 5 of the number 25, the resulting product 150 is greater than the largest number 30 and is divided by all the given numbers without a remainder. This is the smallest possible product (150, 250, 300 ...), which is a multiple of all given numbers.
The numbers 2,3,11,37 are simple, so their LCM is equal to the product of the given numbers.
The rule... To calculate the LCM of prime numbers, you need to multiply all these numbers among themselves.
Another option:
To find the least common multiple (LCM) of several numbers, you need:
1) represent each number as the product of its prime factors, for example:
504 = 2 2 2 3 3 7,
2) write down the powers of all prime factors:
504 = 2 2 2 3 3 7 = 2 3 3 2 7 1,
3) write down all the prime divisors (factors) of each of these numbers;
4) choose the highest degree of each of them, found in all expansions of these numbers;
5) multiply these degrees.
Example... Find the LCM of numbers: 168, 180 and 3024.
Solution... 168 = 2 2 2 3 7 = 2 3 3 1 7 1,
180 = 2 2 3 3 5 = 2 2 3 2 5 1,
3024 = 2 2 2 2 3 3 3 7 = 2 4 3 3 7 1.
We write out the greatest powers of all prime factors and multiply them:
LCM = 2 4 3 3 5 1 7 1 = 15 120.
Common multiples
Simply put, any integer that is divisible by each of the given numbers is common multiple integer data.
You can find the common multiple of two and more whole numbers.
Example 1
Calculate the common multiple of two numbers: $ 2 $ and $ 5 $.
Solution.
By definition, the common multiples of $ 2 $ and $ 5 $ are $ 10 $, since it is a multiple of $ 2 $ and $ 5 $:
The common multiples of the numbers $ 2 $ and $ 5 $ will also be the numbers $ –10, 20, –20, 30, –30 $, etc. they are all divisible by the numbers $ 2 $ and $ 5 $.
Remark 1
Zero is a common multiple of any number of nonzero integers.
According to the properties of divisibility, if a number is a common multiple of several numbers, then the opposite in sign will also be a common multiple of the given numbers. This can be seen from the considered example.
For given integers, you can always find their common multiple.
Example 2
Calculate the common multiple of $ 111 and $ 55.
Solution.
Multiply the given numbers: $ 111 \ div 55 = $ 6105. It is easy to make sure that the number $ 6105 $ is divisible by the number $ 111 $ and by the number $ 55 $:
$ 6105 \ div 111 = $ 55;
$ 6105 \ div 55 = $ 111.
Thus, $ 6105 is the common multiple of $ 111 and $ 55.
Answer: The common multiple of $ 111 $ and $ 55 is $ 6105.
But, as we saw from the previous example, this common multiple is not one. Other common multiples are $ –6105, 12210, –12210, 61050, –61050, etc. Thus, we came to the following conclusion:
Remark 2
Any set of integers has infinitely many common multiples.
In practice, they are limited to finding common multiples of only positive (natural) numbers, since sets of multiples of a given number and its opposite coincide.
Least Common Multiple Determination
The most common multiple of the given numbers is the least common multiple (LCM).
Definition 2
The least positive common multiple of the given integers is least common multiple these numbers.
Example 3
Calculate the LCM of the numbers $ 4 $ and $ 7 $.
Solution.
Because these numbers have no common divisors, then $ LCM (4.7) = $ 28.
Answer: $ LCM (4.7) = $ 28.
Finding the LCM through the GCD
Because there is a relationship between LCM and GCD, with its help you can calculate LCM of two positive numbers :
Remark 3
Example 4
Calculate the LCM of the numbers $ 232 and $ 84.
Solution.
Let's use the formula to find the LCM through the GCD:
$ LCM (a, b) = \ frac (a \ cdot b) (GCD (a, b)) $
Find the GCD of the numbers $ 232 $ and $ 84 $ using Euclid's algorithm:
$ 232 = 84 \ cdot 2 + 64 $,
$ 84 = 64 \ cdot 1 + 20 $,
$ 64 = 20 \ cdot 3 + 4 $,
Those. $ Gcd (232, 84) = $ 4.
Find $ LCM (232, 84) $:
$ LCM (232.84) = \ frac (232 \ cdot 84) (4) = 58 \ cdot 84 = $ 4872
Answer: $ LCM (232.84) = $ 4872.
Example 5
Calculate $ LCM (23, 46) $.
Solution.
Because $ 46 is divisible by $ 23, then $ gcd (23, 46) = $ 23. Find the LCM:
$ LCM (23.46) = \ frac (23 \ cdot 46) (23) = 46 $
Answer: $ LCM (23.46) = $ 46.
Thus, we can formulate the rule:
Remark 4
Online calculator allows you to quickly find the largest common divisor and the least common multiple for both two and any other number of numbers.
Calculator for finding GCD and LCM
Find GCD and LCM
Found GCD and NOC: 5806
How to use the calculator
- Enter numbers in the input field
- If you enter incorrect characters, the input field will be highlighted in red
- click the button "Find GCD and LCM"
How to enter numbers
- Numbers are entered separated by space, period or comma
- The length of the entered numbers is not limited, so finding the GCD and LCM of long numbers will not be difficult
What are GCD and NOC?
Greatest common divisor multiple numbers - this is the largest natural integer by which all original numbers are divisible without a remainder. The greatest common factor is abbreviated as Gcd.
Least common multiple several numbers are smallest number, which is divisible by each of the original numbers without a remainder. The least common multiple is abbreviated as NOC.
How to check that a number is divisible by another number without a remainder?
To find out whether one number is divisible by another without a remainder, you can use some of the divisibility properties of numbers. Then, by combining them, one can check divisibility into some of them and their combinations.
Some signs of divisibility of numbers
1. The criterion for 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 0, 2, 4, 6 or 8, then the number is even, which means it is divisible by 2.
Example: determine if 34938 is divisible by 2.
Solution: look at the last digit: 8 - so the number is divisible by two.
2. The sign of divisibility of a number by 3
A number is divisible by 3 when the sum of its digits is divisible by three. Thus, to determine if 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 is very large, you can repeat the same process again.
Example: determine if 34938 is divisible by 3.
Solution: we count the sum of digits: 3 + 4 + 9 + 3 + 8 = 27.27 is divisible by 3, which means that the number is divisible by three.
3. The 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 34938 is divisible by 5.
Solution: look at the last digit: 8 means the number is NOT divisible by five.
4. The sign of divisibility of a number by 9
This feature is very similar to the divisibility by three: a number is divisible by 9 when the sum of its digits is divisible by 9.
Example: determine if 34938 is divisible by 9.
Solution: we count the sum of digits: 3 + 4 + 9 + 3 + 8 = 27.27 is divisible by 9, which means that the number is divisible by nine.
How to find the gcd and LCM of two numbers
How to find the gcd of two numbers
Most in a simple way calculating the greatest common divisor of two numbers is to find all possible divisors of those numbers and select the largest one.
Let us consider this method using the example of finding the GCD (28, 36):
- Factor both numbers: 28 = 1 2 2 7, 36 = 1 2 2 3 3
- We find the common factors, that is, those that both numbers have: 1, 2 and 2.
- We calculate the product of these factors: 1 · 2 · 2 = 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 least 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 consider only 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), as is already known, is equal to 4
- LCM (28, 36) = 1008/4 = 252.
Finding GCD and LCM for several numbers
The greatest common factor can be found for several numbers, not just two. For this, the numbers to be searched for the greatest common factor 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 relationship is valid for the least common multiple: LCM (a, b, c) = LCM (LCM (a, b), c)
Example: find GCD and LCM for numbers 12, 32 and 36.
- First, factor the numbers: 12 = 1 2 2 3, 32 = 1 2 2 2 2 2 2, 36 = 1 2 2 3 3 3.
- Let's find common factors: 1, 2 and 2.
- Their product will give GCD: 1 2 2 = 4
- Let us now 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 2 3, 36 = 1 2 2 3 3, GCD = 1 2 2 3 = 12.
- LCM (12, 32, 36) = 96 36/12 = 288.
Let's continue talking 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 numbers or more, we will analyze the question of how to find the LCM of a negative number.
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Calculating the least common multiple (LCM) in terms of gcd
We have already established the relationship between the least common multiple and the greatest common divisor. Now we will learn how to determine the LCM in terms of the GCD. Let's first figure out how to do this for positive numbers.
Definition 1
You can find the least common multiple in terms of the greatest common divisor by the formula LCM (a, b) = a b: GCD (a, b).
Example 1
Find the LCM of 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 gcd of numbers 70 and 126. For this we need Euclid's algorithm: 126 = 70 1 + 56, 70 = 56 1 + 14, 56 = 14 4, therefore, GCD (126 , 70) = 14 .
We calculate the LCM: LCM (126, 70) = 126 70: GCD (126, 70) = 126 70: 14 = 630.
Answer: LCM (126, 70) = 630.
Example 2
Find the knock of numbers 68 and 34.
Solution
GCD in this case is not difficult, since 68 is divisible by 34. We 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 of finding the least common multiple for positive integers a and b: if the first number is divisible by the second, 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 factoring numbers into prime factors.
Definition 2
To find the least common multiple, we need to perform a number of simple steps:
- compose the product of all prime factors of the numbers for which we need to find the LCM;
- we exclude all prime factors from the obtained products;
- the product obtained after eliminating common prime factors will be equal to the LCM of these 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 becomes clear: the product of the numbers a and b is equal to the product of all factors that are involved in the decomposition of these two numbers. In this case, the GCD of two numbers is equal to the product all prime factors that are simultaneously present in the factorizations of the given 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 factors of the two original numbers, you get: 2 3 3 5 5 5 7.
If we exclude the factors 3 and 5 common for both numbers, we get a product of the following form: 2 3 5 5 7 = 1050... This product will be our LCM for numbers 75 and 210.
Example 4
Find the LCM of numbers 441 and 700 by expanding 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 decomposition of these numbers will have the form: 2 2 3 3 5 5 7 7 7... Find the common factors. This number is 7. Let's exclude it from the general work: 2 2 3 3 5 5 7 7... It turns out that the 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 the missing factors of the second number to the product of prime factors of the first 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 decompose them into prime factors: 75 = 3 5 5 and 210 = 2 3 5 7... To the product of factors 3, 5 and 5 the number 75 add the missing factors 2 and 7 number 210. We get: 2 · 3 · 5 · 5 · 7. This is the LCM of numbers 75 and 210.
Example 6
Calculate the LCM of numbers 84 and 648.
Solution
Let us 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 the factors 2, 2, 3 and 7
number 84 missing factors 2, 3, 3 and
3
number 648. We get the work 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
Suppose we have integers a 1, a 2,…, a k... NOC 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).
Now let's look at how you can apply the theorem to solve specific problems.
Example 7
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). We apply Euclid's algorithm to calculate the GCD of 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 = 1 260. Therefore, m 2 = 1,260.
Now we calculate by 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 = 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: LCM (140, 9, 54, 250) = 94,500.
As you can see, the calculations are simple, but rather 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 us decompose all five numbers into prime factors: 84 = 2 2 3 7, 6 = 2 3, 48 = 2 2 2 2 2 3, 7, 143 = 11 13. Prime numbers, which is the number 7, cannot be decomposed into prime factors. Such numbers coincide with their prime factorization.
Now take the product of prime factors 2, 2, 3 and 7 of 84 and add the missing factors of the second number to them. We split 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 factors. We pass to the number 48, from the product of prime factors of which we take 2 and 2. Then add a prime factor of 7 of the fourth number and factors of 11 and 13 for 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
To find the least common multiple negative numbers, these numbers must first be replaced by numbers with the opposite sign, and then the calculations must 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 we accept that a and - a- opposite numbers,
then the set of multiples a matches the set of multiples - 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 on opposite numbers 145 and 45 ... Now, according to the algorithm, we calculate the LCM (145, 45) = 145 45: GCD (145, 45) = 145 45: 5 = 1 305, having previously determined the GCD according to 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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