How to find the least common multiple. Finding the least common multiple: methods, examples of finding LCM
But many natural numbers are also divisible by other natural numbers.
For example:
The number 12 is divisible by 1, by 2, by 3, by 4, by 6, by 12;
The number 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 divisible by a whole (for 12 these are 1, 2, 3, 4, 6 and 12) are called divisors of numbers. Divisor of a natural number a- is a natural number that divides a given number a without a trace. A natural number that has more than two divisors is called composite .
Please note that the numbers 12 and 36 have common factors. These numbers are: 1, 2, 3, 4, 6, 12. The greatest divisor of these numbers is 12. The common divisor of these two numbers a And b- this is the number by which both given numbers are divided without remainder a And b.
Common multiples several 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 common multiples there is always a smallest one, in this case it is 90. This number is called the smallestcommon multiple (CMM).
The LCM is always a natural number that must be greater than the largest of the numbers for which it is defined.
Least common multiple (LCM). Properties.
Commutativity:
Associativity:
In particular, if and are coprime numbers, then:
Least common multiple of two integers m And n is a 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 also:
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).
NOC( a, b) can be calculated in several ways:
1. If the greatest common divisor is known, you can use its connection with the LCM:
2. Let the canonical decomposition of both numbers into prime factors be known:
Where p 1 ,...,p k- various prime numbers, and d 1 ,...,d k And e 1 ,...,e k— non-negative integers (they can be zeros if the corresponding prime is not in the expansion).
Then NOC ( a,b) is calculated by the formula:
In other words, the LCM decomposition contains all prime factors included in at least one of the decompositions of numbers a, b, and the largest of the two exponents of this multiplier is taken.
Example:
Calculating the least common multiple of several numbers can be reduced to several sequential calculations of the LCM of two numbers:
Rule. To find the LCM of a series of numbers, you need:
- decompose numbers into prime factors;
- transfer the largest decomposition (the product of the factors of the largest number of the given ones) to the factors of the desired product, and then add factors from the decomposition of other numbers that do not appear in the first number or appear 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 NOC. 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) are supplemented with a factor of 3 (the 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 are supplemented by the factor 5 of the number 25, the resulting product 150 is greater than the largest number 30 and is divisible by all given numbers without a remainder. This is the smallest possible product (150, 250, 300...) that is a multiple of all given numbers.
The numbers 2,3,11,37 are prime numbers, so their LCM is equal to the product of the given numbers.
Rule. To calculate the LCM of prime numbers, you need to multiply all these numbers together.
Another option:
To find the least common multiple (LCM) of several numbers you need:
1) represent each number as a 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 (multipliers) of each of these numbers;
4) choose the greatest degree of each of them, found in all expansions of these numbers;
5) multiply these powers.
Example. Find the LCM of the 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 greatest degrees all prime divisors and multiply them:
NOC = 2 4 3 3 5 1 7 1 = 15120.
Mathematical expressions and problems require a lot of additional knowledge. NOC is one of the main ones, especially often used in The topic is studied in high school, and it is not particularly difficult to understand material; a person familiar with powers and the multiplication table will not have difficulty identifying the necessary numbers and discovering the result.
Definition
A common multiple is a number that can be completely divided into two numbers at the same time (a and b). Most often, this number is obtained by multiplying the original numbers a and b. The number must be divisible by both numbers at once, without deviations.
NOC is the short name adopted for the designation, collected from the first letters.
Ways to get a number
The method of multiplying numbers is not always suitable for finding the LCM; it is much better suited for simple single-digit or two-digit numbers. It is customary to divide into factors; the larger the number, the more factors there will be.
Example #1
For the simplest example, schools usually use prime, single- or double-digit numbers. For example, you need to solve the following task, find the least common multiple of the numbers 7 and 3, the solution is quite simple, just multiply them. As a result, there is a number 21, there is simply no smaller number.
Example No. 2
The second version of the task is much more difficult. The numbers 300 and 1260 are given, finding the LOC is mandatory. To solve the problem, the following actions are assumed:
Decomposition of the first and second numbers into simple factors. 300 = 2 2 * 3 * 5 2 ; 1260 = 2 2 * 3 2 *5 *7. The first stage is completed.
The second stage involves working with already obtained data. Each of the received numbers must participate in the calculation of the final result. For each multiplier, the most large number occurrences. NOC is total number, therefore, the factors from the numbers must be repeated in it, every single one, even those that are present in one copy. Both initial numbers contain the numbers 2, 3 and 5, in different powers; 7 is present only in one case.
To calculate the final result, you need to take each number in the largest of the powers represented into the equation. All that remains is to multiply and get the answer; if filled out correctly, the task fits into two steps without explanation:
1) 300 = 2 2 * 3 * 5 2 ; 1260 = 2 2 * 3 2 *5 *7.
2) NOC = 6300.
That’s the whole problem, if you try to calculate the required number by multiplication, then the answer will definitely not be correct, since 300 * 1260 = 378,000.
Examination:
6300 / 300 = 21 - correct;
6300 / 1260 = 5 - correct.
The correctness of the result obtained is determined by checking - dividing the LCM by both original numbers; if the number is an integer in both cases, then the answer is correct.
What does NOC mean in mathematics?
As you know, there is not a single useless function in mathematics, this one is no exception. The most common purpose of this number is to reduce fractions to common denominator. What is usually studied in grades 5-6 of secondary school. It is also additionally a common divisor for all multiples, if such conditions are present in the problem. Such an expression can find a multiple not only of two numbers, but also of a much larger number - three, five, and so on. How more numbers- the more actions there are in the task, but the complexity does not increase.
For example, given the numbers 250, 600 and 1500, you need to find their common LCM:
1) 250 = 25 * 10 = 5 2 *5 * 2 = 5 3 * 2 - this example describes factorization in detail, without reduction.
2) 600 = 60 * 10 = 3 * 2 3 *5 2 ;
3) 1500 = 15 * 100 = 33 * 5 3 *2 2 ;
In order to compose an expression, it is necessary to mention all the factors, in this case 2, 5, 3 are given - for all these numbers it is necessary to determine the maximum degree.
Attention: all factors must be brought to the point of complete simplification, if possible, decomposed to the level of single digits.
Examination:
1) 3000 / 250 = 12 - correct;
2) 3000 / 600 = 5 - true;
3) 3000 / 1500 = 2 - correct.
This method does not require any tricks or genius level abilities, everything is simple and clear.
Another way
In mathematics, many things are connected, many things can be solved in two or more ways, the same goes for finding the least common multiple, LCM. The following method can be used in the case of simple two-digit and single-digit numbers. A table is compiled into which the multiplicand is entered vertically, the multiplier horizontally, and the product is indicated in the intersecting cells of the column. You can reflect the table using a line, take a number and write down the results of multiplying this number by integers, from 1 to infinity, sometimes 3-5 points are enough, the second and subsequent numbers undergo the same computational process. Everything happens until a common multiple is found.
Given the numbers 30, 35, 42, you need to find the LCM connecting all the numbers:
1) Multiples of 30: 60, 90, 120, 150, 180, 210, 250, etc.
2) Multiples of 35: 70, 105, 140, 175, 210, 245, etc.
3) Multiples of 42: 84, 126, 168, 210, 252, etc.
It is noticeable that all the numbers are quite different, the only common number among them is 210, so it will be the NOC. Among the processes associated with this calculation there is also the largest common divisor, which is calculated according to similar principles and is often found in neighboring problems. The difference is small, but quite significant; LCM involves calculating a number that is divisible by all data initial values, and GCD involves the calculation highest value by which the original numbers are divided.
Definition. The largest 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 greatest common divisor of the 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.
Factoring the numbers 48 and 36, we 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 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 factor 75 and 60 into prime factors: 75 = 3 * 5 * 5, and 60 = 2 * 2 * 3 * 5.
Let's write down 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 is due to the fact that any number is either prime or can be represented as a product of 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 every 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 online calculator allows you to quickly find the greatest common divisor and least common multiple for two or any other number of numbers.
Calculator for finding GCD and LCM
Find GCD and LOC
Found GCD and LOC: 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 "Find GCD and LOC" button
How to enter numbers
- Numbers are entered separated by a space, period or comma
- The length of entered numbers is not limited, so finding GCD and LCM of long numbers is not difficult
What are GCD and NOC?
Greatest common divisor several numbers is the largest natural integer by which all 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 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 properties of divisibility of numbers. Then, by combining them, you can check the divisibility of some of them and their combinations.
Some signs of divisibility of numbers
1. Divisibility test for 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 whether the number 34938 is divisible by 2.
Solution: look at the last digit: 8 means the number is divisible by two.
2. Divisibility test for a number by 3
A number is divisible by 3 when the sum of its digits is divisible by three. Thus, to determine whether a number is divisible by 3, you need to calculate the sum of the digits and check whether it is divisible by 3. Even if the sum of the digits is very large, you can repeat the same process again.
Example: determine whether the number 34938 is divisible by 3.
Solution: We count the sum of the numbers: 3+4+9+3+8 = 27. 27 is divisible by 3, which means the number is divisible by three.
3. Divisibility test for a number by 5
A number is divisible by 5 when its last digit is zero or five.
Example: determine whether the number 34938 is divisible by 5.
Solution: look at the last digit: 8 means the number is NOT divisible by five.
4. Divisibility test for 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 whether the number 34938 is divisible by 9.
Solution: We count the sum of the numbers: 3+4+9+3+8 = 27. 27 is divisible by 9, which means the number is divisible by nine.
How to find 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 these numbers and select the largest of them.
Let's consider this method using the example of finding GCD(28, 36):
- We factor 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 = 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 method is that you can write down the first multiples of two numbers, and then choose among them 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 numbers 28 and 36: 28·36 = 1008
- GCD(28, 36), as already known, is equal to 4
- LCM(28, 36) = 1008 / 4 = 252 .
Finding GCD and LCM for several numbers
The greatest common divisor can be found for several numbers, not just two. To do 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. You can also use the following relation to find the gcd of several numbers: GCD(a, b, c) = GCD(GCD(a, b), c).
A similar relationship applies to 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, let's factor the numbers: 12 = 1·2·2·3, 32 = 1·2·2·2·2·2, 36 = 1·2·2·3·3.
- Let's find the common factors: 1, 2 and 2.
- Their product will give GCD: 1·2·2 = 4
- Now let’s find the LCM: to do this, let’s first find the LCM(12, 32): 12·32 / 4 = 96 .
- To find the NOC of everyone three numbers, you need to find 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.
Common multiples
Simply put, any integer that is divisible by each of the given numbers is common multiple given integers.
You can find the common multiple of two and more integers.
Example 1
Calculate the common multiple of two numbers: $2$ and $5$.
Solution.
By definition, the common multiple of $2$ and $5$ is $10$, because it is a multiple of the number $2$ and the number $5$:
Common multiples of the numbers $2$ and $5$ will also be the numbers $–10, 20, –20, 30, –30$, etc., because they are all divided into numbers $2$ and $5$.
Note 1
Zero is a common multiple of any number of non-zero integers.
According to the properties of divisibility, if a certain number is a common multiple of several numbers, then the number opposite in sign will also be a common multiple of the given numbers. This can be seen from the example considered.
For given integers, you can always find their common multiple.
Example 2
Calculate the common multiple of $111$ and $55$.
Solution.
Let's multiply the given numbers: $111\div 55=6105$. It is easy to verify that the number $6105$ is divisible by the number $111$ and the number $55$:
$6105\div 111=$55;
$6105\div 55=$111.
Thus, $6105$ is a common multiple of $111$ and $55$.
Answer: The common multiple of $111$ and $55$ is $6105$.
But, as we have already seen from the previous example, this common multiple is not one. Other common multiples would be $–6105, 12210, –12210, 61050, –61050$, etc. Thus, we came to the following conclusion:
Note 2
Any set of integers has an infinite number of common multiples.
In practice, they are limited to finding common multiples of only positive integer (natural) numbers, because the sets of multiples of a given number and its opposite coincide.
Determining Least Common Multiple
Of all multiples of given numbers, the least common multiple (LCM) is used most often.
Definition 2
The least positive common multiple of 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: $NOK (4,7)=28$.
Finding NOC via GCD
Because there is a connection between LCM and GCD, with its help you can calculate LCM of two positive integers:
Note 3
Example 4
Calculate the LCM of the numbers $232$ and $84$.
Solution.
Let's use the formula to find the LCM through GCD:
$LCD (a,b)=\frac(a\cdot b)(GCD (a,b))$
Let's find the GCD of the numbers $232$ and $84$ using the Euclidean algorithm:
$232=84\cdot 2+64$,
$84=64\cdot 1+20$,
$64=20\cdot 3+4$,
Those. $GCD(232, 84)=4$.
Let's find $LCC (232, 84)$:
$NOK (232.84)=\frac(232\cdot 84)(4)=58\cdot 84=4872$
Answer: $NOK (232.84)=$4872.
Example 5
Compute $LCD(23, 46)$.
Solution.
Because $46$ is divisible by $23$, then $gcd (23, 46)=23$. Let's find the LOC:
$NOK (23.46)=\frac(23\cdot 46)(23)=46$
Answer: $NOK (23.46)=$46.
Thus, one can formulate rule:
Note 4
