How to find the least common multiple? Divisors and multiples.

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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

How to use the calculator

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
The length of the entered numbers is not limited, so finding the gcd and lcm of long numbers will not be difficult

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.
Decision: 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.
Decision: 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.
Decision: 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.
Decision: 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

Most in a simple way calculating the greatest common divisor of two numbers is to find all possible divisors of those 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 searched 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 .

Students are given a lot of math assignments. Among them, very often there are tasks with the following formulation: there are two values. How to find the least common multiple of given numbers? It is necessary to be able to perform such tasks, since the acquired skills are used to work with fractions when different denominators. In the article, we will analyze how to find the LCM and the basic concepts.

Before finding the answer to the question of how to find the LCM, you need to define the term multiple. Most often, the wording of this concept is as follows: a multiple of some value A is a natural number that will be divisible by A without a remainder. So, for 4, 8, 12, 16, 20 and so on, up to the necessary limit.

In this case, the number of divisors for a particular value can be limited, and there are infinitely many multiples. There is also the same value for natural values. This is an indicator that is divided by them without a remainder. Having dealt with the concept of the smallest value for certain indicators, let's move on to how to find it.

Finding the NOC

The least multiple of two or more exponents is the smallest natural number that is fully divisible by all the given numbers.

There are several ways to find such a value. Let's consider the following methods:

If the numbers are small, then write in the line all divisible by it. Keep doing this until you find something in common among them. In the record, they are denoted by the letter K. For example, for 4 and 3, the smallest multiple is 12.
If these are large or you need to find a multiple for 3 or more values, then you should use a different technique here, which involves decomposing numbers into prime factors. First, lay out the largest of the indicated, then all the rest. Each of them has its own number of multipliers. As an example, let's decompose 20 (2*2*5) and 50 (5*5*2). For the smaller of them, underline the factors and add to the largest. The result will be 100, which will be the least common multiple of the above numbers.
When finding 3 numbers (16, 24 and 36) the principles are the same as for the other two. Let's expand each of them: 16 = 2*2*2*2, 24=2*2*2*3, 36=2*2*3*3. Only two deuces from the expansion of the number 16 were not included in the decomposition of the largest. We add them and get 144, which is the smallest result for the previously indicated numerical values.

Now we know what is the general technique for finding the smallest value for two, three or more values. However, there are also private methods, helping to search for NOCs, if the previous ones do not help.

How to find GCD and NOC.

Private Ways of Finding

As with any mathematical section, there are special cases of finding LCMs that help in specific situations:

if one of the numbers is divisible by the others without a remainder, then the lowest multiple of these numbers is equal to it (NOC 60 and 15 is equal to 15);
mutually prime numbers do not have common prime divisors. Their smallest value is equal to the product of these numbers. Thus, for the numbers 7 and 8, this will be 56;
the same rule works for other cases, including special ones, which can be read about in specialized literature. This should also include cases of decomposition of composite numbers, which are the subject of separate articles and even Ph.D. dissertations.

Special cases are less common than standard examples. But thanks to them, you can learn how to work with fractions of varying degrees of complexity. This is especially true for fractions., where there are different denominators.

Some examples

Let's look at a few examples, thanks to which you can understand the principle of finding the smallest multiple:

We find LCM (35; 40). We lay out first 35 = 5*7, then 40 = 5*8. Add to lowest digit 8 and get NOK 280.
NOC (45; 54). We lay out each of them: 45 = 3*3*5 and 54 = 3*3*6. We add the number 6 to 45. We get the NOC equal to 270.
Well, the last example. There are 5 and 4. There are no simple multiples for them, so the least common multiple in this case will be their product, equal to 20.

Thanks to examples, you can understand how the NOC is located, what are the nuances and what is the meaning of such manipulations.

Finding the NOC is much easier than it might seem at first. For this, both a simple decomposition and the multiplication of simple values \u200b\u200bto each other are used.. The ability to work with this section of mathematics helps in the further study of mathematical topics, especially fractions of varying degrees of complexity.

Do not forget to periodically solve examples various methods, this develops the logical apparatus and allows you to remember numerous terms. Learn methods for finding such an indicator and you will be able to work well with the rest of the mathematical sections. Happy learning math!

Video

This video will help you understand and remember how to find the least common multiple.

How to find the least common multiple?

It is necessary to find each factor of each of the two numbers for which we find the least common multiple, and then multiply the factors that coincided with the first and second numbers by each other. The result of the product will be the desired multiple.

For example, we have the numbers 3 and 5 and we need to find the LCM (least common multiple). Us must be multiplied and three and five for all numbers starting from 1 2 3 ... and so on until we see the same number here and there.

We multiply the three and get: 3, 6, 9, 12, 15

Multiply five and get: 5, 10, 15

The prime factorization method is the most classic for finding the least common multiple (LCM) of multiple numbers. This method is clearly and simply demonstrated in the following video:

Adding, multiplying, dividing, reducing to a common denominator and other arithmetic operations is a very exciting activity, examples that occupy a whole sheet are especially admired.

So find the common multiple for two numbers, which will be the smallest number by which two numbers are divisible. I want to note that it is not necessary to resort to formulas in the future to find what you are looking for, if you can count in your mind (and this can be trained), then the numbers themselves pop up in your head and then the fractions click like nuts.

To begin with, we will learn that we can multiply two numbers against each other, and then reduce this figure and divide alternately by these two numbers, so we will find the smallest multiple.

For example, two numbers 15 and 6. We multiply and get 90. This is clearly a larger number. Moreover, 15 is divisible by 3 and 6 is divisible by 3, which means we also divide 90 by 3. We get 30. We try to divide 30 by 15 is 2. And 30 divides 6 is 5. Since 2 is the limit, it turns out that the smallest multiple for the numbers 15 and 6 will be 30.

With more numbers it will be a little more difficult. but if you know which numbers give a zero remainder when divided or multiplied, then, in principle, there are no big difficulties.

How to find the NOC

Here is a video that will show you two ways to find the least common multiple (LCM). By practicing using the first of the proposed methods, you can better understand what the least common multiple is.

Here's another way to find the least common multiple. Let's take a look at an illustrative example.

It is necessary to find the LCM of three numbers at once: 16, 20 and 28.

We represent each number as the product of its prime factors:

We write down the powers of all prime factors:

16 = 224 = 2^24^1

20 = 225 = 2^25^1

28 = 227 = 2^27^1

We select all prime divisors (multipliers) with the largest degrees, multiply them and find the LCM:

LCM = 2^24^15^17^1 = 4457 = 560.

LCM(16, 20, 28) = 560.

Thus, as a result of the calculation, the number 560 was obtained. It is the least common multiple, that is, it is divisible by each of the three numbers without a remainder.

The least common multiple is the number that can be divided by several given numbers without a remainder. In order to calculate such a figure, you need to take each number and decompose it into simple factors. Those numbers that match are removed. Leaves everyone one at a time, multiply them among themselves in turn and get the desired - the least common multiple.

NOC, or least common multiple, is the smallest natural number of two or more numbers that is divisible by each of the given numbers without a remainder.

Here is an example of how to find the least common multiple of 30 and 42.

The first step is to decompose these numbers into prime factors.

For 30, it's 2 x 3 x 5.

For 42, this is 2 x 3 x 7. Since 2 and 3 are in the expansion of the number 30, we cross them out.

We write out the factors that are included in the expansion of the number 30. This is 2 x 3 x 5.
Now you need to multiply them by the missing factor, which we have when decomposing 42, and this is 7. We get 2 x 3 x 5 x 7.
We find what is equal to 2 x 3 x 5 x 7 and get 210.

As a result, we get that the LCM of the numbers 30 and 42 is 210.

To find the least common multiple, you need to follow a few simple steps in sequence. Consider this using the example of two numbers: 8 and 12

We decompose both numbers into prime factors: 8=2*2*2 and 12=3*2*2
We reduce the same multipliers for one of the numbers. In our case, 2 * 2 match, we reduce them for the number 12, then 12 will have one factor: 3.
Find the product of all remaining factors: 2*2*2*3=24

Checking, we make sure that 24 is divisible by both 8 and 12, and this is the smallest natural number that is divisible by each of these numbers. Here we are find the least common multiple.

I'll try to explain using the example of the numbers 6 and 8. The least common multiple is the number that can be divided by these numbers (in our case, 6 and 8) and there will be no remainder.

So, we begin to multiply first 6 by 1, 2, 3, etc. and 8 by 1, 2, 3, etc.

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 $60$.

Decision:

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 given set will be the number $12$. So the greatest common divisor of $48$ and $60$ is $12$.

Definition of NOC

Definition 3

common multiple 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 $D(a;b)$

Mathematical expressions and tasks require a lot of additional knowledge. NOC is one of the main ones, especially often used in the topic. The topic is studied in high school, while it is not particularly difficult to understand material, it will not be difficult for a person familiar with powers and the multiplication table to single out necessary numbers and discover 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 accepted term for short title, assembled from the first letters.

Ways to get a number

To find the LCM, the method of multiplying numbers is not always suitable, it is much better suited for simple one-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 take simple, one-digit or two-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 the number 21, there is simply no smaller number.

Example #2

The second option is much more difficult. The numbers 300 and 1260 are given, finding the LCM is mandatory. To solve the task, the following actions are assumed:

Decomposition of the first and second numbers into the simplest factors. 300 = 2 2 * 3 * 5 2 ; 1260 = 2 2 * 3 2 * 5 * 7. The first stage has been completed.

The second stage involves working with the already obtained data. Each of the received numbers must participate in the calculation final result. For each multiplier, the most big number occurrences. NOC is total number, so the factors from the numbers should be repeated in it to the last, even those that are present in one copy. Both initial numbers have in their composition the numbers 2, 3 and 5, in different degrees, 7 is only in one case.

To calculate the final result, you need to take each number in the largest of their represented powers, into the equation. It remains only to multiply and get the answer, with correct filling The task fits into two steps without explanation:

1) 300 = 2 2 * 3 * 5 2 ; 1260 = 2 2 * 3 2 *5 *7.

2) NOK = 6300.

That's the whole task, if you try to calculate the desired number by multiplying, then the answer will definitely not be correct, since 300 * 1260 = 378,000.

Examination:

6300 / 300 = 21 - true;

6300 / 1260 = 5 is correct.

The correctness of the result 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 bring fractions to a common denominator. What is usually studied in grades 5-6 high school. It is also additionally a common divisor for all multiples, if such conditions are in the problem. Such an expression can find a multiple not only of two numbers, but also of much more- three, five and so on. The more numbers - the more actions in the task, but the complexity of this does not increase.

For example, given the numbers 250, 600 and 1500, you need to find their total LCM:

1) 250 = 25 * 10 = 5 2 * 5 * 2 = 5 3 * 2 - this example describes the 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 required to mention all factors, in this case 2, 5, 3 are given - for all these numbers it is required to determine the maximum degree.

Attention: all multipliers must be brought to full simplification, if possible, decomposing to the level of single digits.

Examination:

1) 3000 / 250 = 12 - true;

2) 3000 / 600 = 5 - true;

3) 3000 / 1500 = 2 is correct.

This method does not require any tricks or genius level abilities, everything is simple and clear.

Another way

In mathematics, a lot is connected, a lot 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 digits. A table is compiled in which the multiplier is entered vertically, the multiplier horizontally, and the product is indicated in the intersecting cells of the column. You can reflect the table by means of a line, a number is taken and the results of multiplying this number by integers are written in a row, from 1 to infinity, sometimes 3-5 points are enough, the second and subsequent numbers are subjected to the same computational process. Everything happens until a common multiple is found.

Given the numbers 30, 35, 42, you need to find the LCM that connects 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 LCM. Among the processes associated with this calculation, there is also the greatest common divisor, which is calculated according to similar principles and is often encountered in neighboring problems. The difference is small, but significant enough, the LCM involves the calculation of a number that is divisible by all given initial values, and the GCM involves the calculation greatest value by which the original numbers are divisible.