How to find NOC (the smallest total multiple)

The total multiple for two integers is such an integer that is divided by a focus without a balance on both specified numbers.

The smallest total multiple for two integers is the smallest of all integers, which is divided and without a balance on both specified numbers.

Method 1. It is possible to find the NOK, in turn, for each of the specified numbers, writing out in the order of increasing all the numbers that are obtained by multiplying them by 1, 2, 3, 4, and so on.

Example For numbers 6 and 9.
Multiply the number 6, sequentially, 1, 2, 3, 4, 5.
We get: 6, 12, 18 , 24, 30
We multiply the number 9, sequentially, 1, 2, 3, 4, 5.
We get: 9, 18 , 27, 36, 45
As can be seen, the NOC for numbers 6 and 9 will be equal to 18.

This method is convenient when both numbers are small and easily multiplied by the sequence of integers. However, there are cases when it is necessary to find NOCs for double-digit or three-digit numbers, as well as when the initial numbers are three or even more.

Method 2.. It is possible to find the NOC, spreading the initial numbers to simple factors.
After decomposition, it is necessary to delete the same numbers from the resulting series of simple factors. The remaining numbers of the first number will be a multiplier for the second, and the remaining numbers of the second - a multiplier for the first.

Examplefor number 75 and 60.
The smallest overall multiple numbers 75 and 60 can be found and not prescribing in a row to these numbers. To do this, lay 75 and 60 to simple multipliers:
75 = 3 * 5 * 5, and
60 = 2 * 2 * 3 * 5 .
As can be seen, multipliers 3 and 5 are found in both lines. Mentally, they are "crushing".
Drink out the remaining multipliers in the decomposition of each of these numbers. With the decomposition of the number 75, we left the number 5, and with the decomposition of the number 60 - 2 * 2 remained
It means to determine the NOC for numbers 75 and 60, we need the remaining numbers from decomposition 75 (this is 5) multiply by 60, and the numbers remaining from the decomposition of the number 60 (this is 2 * 2) multiply by 75. That is, for ease of understanding , we say that we multiply "nest".
75 * 2 * 2 = 300
60 * 5 = 300
Thus, we found the NOC for numbers 60 and 75. This is the number 300.

Example. Determine the NOC for numbers 12, 16, 24
In this case, our actions will be somewhat more complicated. But first, as always, we will define all the numbers for simple factors.
12 = 2 * 2 * 3
16 = 2 * 2 * 2 * 2
24 = 2 * 2 * 2 * 3
In order to correctly define the NOC, select the smallest of all numbers (this is the number 12) and consistently pass according to its factor, crossing them, if at least one of the other numbers met the same, not yet stressed multiplier.

Step 1 . We see that 2 * 2 are found in all rows of numbers. Crouch them.
12 = 2 * 2 * 3
16 = 2 * 2 * 2 * 2
24 = 2 * 2 * 2 * 3

Step 2. In ordinary multipliers of the number 12, there is only a number 3. But it is present in simple multipliers of the number 24. Explore the number 3 of both rows, and no action is expected for the number 16.
12 = 2 * 2 * 3
16 = 2 * 2 * 2 * 2
24 = 2 * 2 * 2 * 3

As we see, with the decomposition of the number 12, we "crossed out" all numbers. So the finding of the NOC is completed. It remains only to calculate its value.
For the number 12, we take the remaining multipliers in the number 16 (nearest ascending)
12 * 2 * 2 = 48
It's a nok

As you can see, in this case, the finding of the NOC was somewhat more complicated, but when it is necessary to find it for three or more numbers, this method allows you to make it faster. However, both ways to find NOC are correct.

The greatest common divisor and the smallest general multiple are key arithmetic concepts that allow without effort to operate with ordinary fractions. NOC and most often used to search for a common denominator of several fractions.

Basic concepts

An integer divider X is another integer y, which X is divided without a residue. For example, divider 4 is 2, and 36 - 4, 6, 9. A multiple of the whole X is such a number Y, which is divided into x without a residue. For example, 3 times 15, and 6 - 12.

For any pair of numbers, we can find their common dividers and multiple. For example, for 6 and 9, the total multiple is 18, and a common divider - 3. It is obvious that dividers and multiple pairs can be somewhat, therefore, during the calculations, the largest node divider and the smallest multiple nok are used.

The smallest divider does not make sense, since for any number it is always a unit. The greatest multiple is also meaningless, since the sequence of multiples rushes into infinity.

Finding Node

To search for the greatest common divisor, there are many methods, the most famous of which:

• sequential bust of dividers, the choice of common to the pair and the search for the greatest of them;
• decomposition of numbers for indivisible factors;
• algorithm Euclida;
• binary algorithm.

Today in educational institutions are the most popular methods of decomposition on simple multipliers and the Euclide algorithm. The latter in turn is used in solving diophantine equations: Node search is required to test the equation to the ability to resolve in integers.

Nok.

The smallest total multiple is also determined by consistent bustling or decomposition of indivisible multipliers. In addition, it is easy to find NOC, if the largest divider is already defined. For numbers x and y, NOC and NOD are connected by the following ratio:

NOK (x, y) \u003d x × y / node (x, y).

For example, if NOD (15.18) \u003d 3, then NOK (15.18) \u003d 15 × 18/3 \u003d 90. The most obvious example of the use of the NOC is the search for a common denominator, which is the smallest common multiple for the fractions given.

Mutually simple numbers

If the pair of numbers do not have common divisors, then such a couple is called mutually simple. The node for such pairs is always equal to one, and based on the connection of dividers and multiple, NOCs for mutually simple is equal to their work. For example, the numbers 25 and 28 are mutually simple, because they do not have common divisors, and NOK (25, 28) \u003d 700, which corresponds to their work. Two any indivisible numbers will always be mutually simple.

Calculator of the general divider and multiple

With our calculator, you can calculate NOD and NIC for an arbitrary number of numbers to choose from. The tasks for the calculation of common divisors and multiple are found in arithmetic 5, grade 6, but NOD and NOC are the key concepts of mathematics and are used in the theory of numbers, planimetry and communicative algebra.

Examples from real life

Common denominator fractions

The smallest total is used when searching for a common denominator of several fractions. Suppose in the arithmetic task you need to summarize 5 fractions:

1/8 + 1/9 + 1/12 + 1/15 + 1/18.

To add fractions, the expression must be brought to a common denominator, which comes down to the task of finding the NOC. To do this, select the 5 numbers in the calculator and enter the values \u200b\u200bof the denominators to the corresponding cells. The program will calculate the NOC (8, 9, 12, 15, 18) \u003d 360. Now it is necessary to calculate additional multipliers for each fraction, which are defined as the ratio of the NOC to the denominator. Thus, additional multipliers will look like:

• 360/8 = 45
• 360/9 = 40
• 360/12 = 30
• 360/15 = 24
• 360/18 = 20.

After that, we multiply all the fractions on the corresponding additional factor and get:

45/360 + 40/360 + 30/360 + 24/360 + 20/360.

We can easily summarize such fractions and get the result in the form of 159/360. We reduce the fraction of 3 and see the final answer - 53/120.

Solution of linear diophantic equations

Linear diophanty equations are an expressions of the form AX + BY \u003d D. If the ratio D / Node (A, B) is an integer, the equation is solvable in integers. Let's check a pair of equations for an integer solution. First, check the equation 150x + 8y \u003d 37. With the help of the calculator we find a node (150.8) \u003d 2. Delim 37/2 \u003d 18.5. The number is not integer, therefore, the equation has no integer roots.

We check the equation 1320x + 1760y \u003d 10120. We use a calculator to find a node (1320, 1760) \u003d 440. We divide 10120/440 \u003d 23. As a result, we obtain an integer, therefore, the diophanty equation is solvable in the entire coefficients.

Conclusion

Nodes and NOCs play a large role in the theory of numbers, and the concepts themselves are widely used in various fields of mathematics. Use our calculator to calculate the greatest divisors and the smallest multiple of any number of numbers.

How to find the smallest common multiple?

It is necessary to find every multiplier of each of the two numbers, which we find the smallest common multiple, and then multiply the factors that coincided at the first and second number. The result of the work will be the desired multiple.

For example, we have numbers 3 and 5 and we need to find the NOC (the smallest common multiple). Us we must multiply and triple and praq all numbers starting from 1 2 3 ... And so until we see the same number and there.

Troika and get: 3, 6, 9, 12, 15

Multiply now and get: 5, 10, 15

The decomposition method for simple factors is the most classic to find the smallest common multiple (NOK) for several numbers. Visually and simply demonstrated this method in the next video:

To fold, multiply, divide, lead to a general denominator and other arithmetic actions a very exciting occupation, especially admire examples that occupy a whole sheet.

So find a common multiple for two numbers, which will be the smallest number on which two numbers are divided. I want to note that it is not necessary to continue to resort to the formulas to find the desired if you can count in the mind (and this can be trained), then the numbers themselves pop up in the head and then the fractions are clicked like nuts.

To begin with, I will absorb that you can multiply two numbers on each other, and then reduce this figure and divide alternately for these two numbers, so we find the smallest multiple.

For example, two numbers 15 and 6. Multiply and get 90. This is clearly more than the number. Moreover, it is divided into 3 and 6 divided by 3, which means 90, too, divide by 3. Take 30. We try 30 to divide 15 equals 2. and 30 divide 6 is 5. Since 2 is a limit, it turns out that the smallest multiple for numbers 15 and 6 will be 30.

With the numbers more will be a little more difficult. But if you know what numbers give a zero residue during division or multiplication, then difficulties, in principle, are not large.

• How to find nook

  Here is a video in which you will be offered two ways to find the smallest common multiple (NOC). Disadvantaged to use the first of the proposed methods, you can better understand what the smallest is the least multiple.

I present another way to find the smallest common multiple. Consider it on a visual example.

It is necessary to find the NOK at once the TRX numbers: 16, 20 and 28.

• We present every number as a product of its simple factors:
• We write down the degrees of all simple multipliers:

16 = 224 = 2^24^1

20 = 225 = 2^25^1

28 = 227 = 2^27^1

• We choose all simple dividers (multipliers) with the highest degrees, we turn them out and find the NOC:

Nok \u003d 2 ^ 24 ^ 15 ^ 17 ^ 1 \u003d 4457 \u003d 560.

NOK (16, 20, 28) \u003d 560.

Thus, as a result, the calculation turned out the number 560. It is the lowest common multiple, that is, it is divided into each of the three numbers without a residue.

The smallest total multiple number is such a figure that is divided into several proposed numbers without a residue. In order for such a digit to calculate, you need to take each number and decompose it on simple factors. Those numbers that match, remove. It leaves everyone alone, turn them together in turn and we get the desired - the smallest common pain.

Nok, or the smallest common pain- This is the smallest natural number of two or more numbers, which is divided into each of the data numbers without a residue.

Here is an example of how to find the smallest total multiple 30 and 42.

• First of all, you need to decompose the number of numbers on simple factors.

For 30, it is 2 x 3 x 5.

For 42, it is 2 x 3 x 7. Since 2 and 3 are in the decomposition of the number 30, then strike them.

• We write out multipliers that are included in the decomposition of the number 30. These are 2 x 3 x 5.
• Now you need to draw them to the missing multiplier, which we have in decomposition 42, and this is 7. We obtain 2 x 3 x 5 x 7.
• We find what is 2 x 3 x 5 x 7 and we get 210.

As a result, we obtain that the NOC numbers 30 and 42 are 210.

To find the smallest total multipleYou need to perform successively slightly simple actions. Consider this on the example of two numbers: 8 and 12

1. Decompose both numbers on simple multipliers: 8 \u003d 2 * 2 * 2 and 12 \u003d 3 * 2 * 2
2. We reduce the same multipliers from one of the numbers. In our case, 2 * 2 coincide, reduce them for a number 12, then 12 will remain one multiplier: 3.
3. We find the work of all the remaining multipliers: 2 * 2 * 2 * 3 \u003d 24

Checking, we are convinced that 24 is divided into 8 and by 12, and this is the smallest natural number that is divided into each of these numbers. Here we are I. found the smallest total multiple.

I will try to explain on the example of numbers 6 and 8. The smallest common multiple is the number that can be divided into these numbers (in our case 6 and 8) and the residue will not.

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

The smallest total multiple two numbers is directly related to the greatest common divisor of these numbers. This communication between Nod and Nok Determined by the following theorem.

Theorem.

The smallest total multiple of two positive integers A and B is equal to the product of the numbers a and b, divided into the largest common divisor of numbers a and b, that is, NOK (A, B) \u003d A · B: Node (A, B).

Evidence.

Let M is some multiple numbers a and b. That is, M is divided into A, and by definition of divisibility there exists some integer K such that the equality m \u003d a · k is true. But m is divided into B, then A · K is divided into b.

Denote by Node (A, B) as d. Then you can record the equalities a \u003d a 1 · d and b \u003d b 1 · d, with 1 \u003d a: d and b 1 \u003d b: D will be mutually simple numbers. Consequently, the condition obtained in the previous paragraph that A · K is divided into B, it is possible to reformulate as follows: a 1 · d · k is divided into B 1 · d, and this is due to the properties of divisibility equivalent to the condition that A 1 · k is divided into b one .

You also need to record two important consequences of the theorem considered.

Common multiple two numbers coincide with multiple of their smallest common paint.

This is true, since any common multiple M numbers A and B is determined by the equality m \u003d nok (a, b) · t with some whole value t.

The smallest common multiple of mutually simple positive numbers A and B are equal to their work.

The rationale for this fact is quite obvious. Since A and B are mutually simple, then node (a, b) \u003d 1, therefore, NOK (A, B) \u003d A · B: Node (A, B) \u003d A · B: 1 \u003d A · B.

The smallest total multiple of three and more numbers

Finding the smallest total multiple of three and more numbers can be reduced to the sequential finding of the NOC of the two numbers. As is done, indicated in the following theorem.a 1, a 2, ..., a k coincide with the common multiple numbers M k - 1 and a k, therefore, coincide with multiple numbers M K. And since the smallest positive multiple number M K is the number M K, the smallest common multiple numbers A 1, A 2, ..., A K is m k.

List of references.

• Vilenkin N.Ya. and others. mathematics. Grade 6: Textbook for general educational institutions.
• Vinogradov I.M. Fundamentals of the theory of numbers.
• Mikhelovich Shh. The theory of numbers.
• Kulikov L.Ya. and others. Collection of tasks on algebra and theory of numbers: Tutorial for students Fiz.-Mat. specialties of pedagogical institutions.

Online Calculator allows you to quickly find the largest common divider and the smallest common to both for two and for any other number of numbers.

Calculator for finding nodes and nok

Find node and nok

Node and Nok are found: 5806

How to use the calculator

• Enter the numbers in the input field
• In the case of input incorrect characters, the input box will be highlighted in red
• click "Find Node and Nok"

How to enter numbers

• The numbers are introduced through a space, point or comma
• The length of the input numbers is not limited.so finding nodes and nok long numbers will not be difficult

What is NOD and NOK?

The greatest common divisel There are several numbers - this is the largest natural integer on which all initial numbers are divided without a residue. The greatest common divisor is abbreviated as Node.
The smallest common pain There are several numbers - this is the smallest number that is divided into each of the initial numbers without a residue. The smallest common multiple is written abbreviated as Nok..

How to check that the number is divided into another number without a residue?

To find out if one number is divided into another without a residue, you can use some properties of the divisibility of numbers. Then, combining them, you can check the divisibility on some of them and their combinations.

Some signs of the divisibility of numbers

1. Sign of the divisibility of the number by 2
To determine whether the number is divided into two (whether it is even used), just look at the last figure of this number: if it is equal to 0, 2, 4, 6 or 8, then the number is clearly, which means it is divided by 2.
Example: Determine whether it is divided by 2 number 34938.
Decision: We look at the last digit: 8 means the number is divided into two.

2. Sign of the divisibility of the number by 3
The number is divided by 3 when the sum of its numbers is divided into three. Thus, to determine if the number is divided into 3, it is necessary to calculate the amount of numbers and check whether it is divided by 3. Even if the amount of numbers turned out to be very large, you can repeat the same process again.
Example: Determine whether the number 34938 is divided into 3.
Decision: We consider the amount of numbers: 3 + 4 + 9 + 3 + 8 \u003d 27. 27 is divided into 3, and therefore the number is divided into three.

3. Sign of the divisibility of the number on 5
The number is divided by 5 when its last digit is zero or five.
Example: Determine whether the number 34938 is divided into 5.
Decision: We look at the last digit: 8 means the number is not divided by five.

4. Sign of the divisibility of the number by 9
This feature is very similar to a sign of divisibility on the top: the number is divided by 9 when the amount of its numbers is divided into 9.
Example: Determine whether the number 34938 is divided into 9.
Decision: We consider the amount of numbers: 3 + 4 + 9 + 3 + 8 \u003d 27. 27 is divided into 9, and therefore the number is divided by nine.

How to find nodes and nok two numbers

How to find a node two numbers

The simplest way to calculate the greatest general divisor of two numbers is to search for all possible divisors of these numbers and choosing the greatest of them.

Consider this method on the example of finding Node (28, 36):

1. Obtained both numbers on multipliers: 28 \u003d 1 · 2 · 2 · 7, 36 \u003d 1 · 2 · 2 · 3 · 3
2. We find general multipliers, that is, those that have both numbers: 1, 2 and 2.
3. Calculate the product of these multipliers: 1 · 2 · 2 \u003d 4 - this is the greatest common divisor of numbers 28 and 36.

How to find a nok two numbers

The most common two ways to find the smallest multiple two numbers are most common. The first way is that it is possible to write down the first multiple two numbers, and then choose among them an such number that will be common to both numbers and at the same time. And the second is to find the node of these numbers. Consider only it.

To calculate the NOC, it is necessary to calculate the product of the initial numbers and then divide it into a pre-found node. Find the NOC for the same numbers 28 and 36:

1. We find the product of numbers 28 and 36: 28 · 36 \u003d 1008
2. Node (28, 36), as already known, equal to 4
3. NOK (28, 36) \u003d 1008/4 \u003d 252.

Finding node and nok for several numbers

The largest shared divider can be found for several numbers, and not just for two. For this purpose, the number to be searched for the greatest common divisor is unfolded on simple factors, then a product of common simple multipliers of these numbers are found. Also for finding a node of several numbers, you can use the following ratio: Node (a, b, c) \u003d node (node \u200b\u200b(a, b), c).

A similar relation is valid for the smallest common multiple numbers: NOK (A, B, C) \u003d NOC (NOK (A, B), C)

Example: Find Nodes and Nok for numbers 12, 32 and 36.

1. The captured the numbers on the multipliers: 12 \u003d 1 · 2 · 2 · 3, 32 \u003d 1 · 2 · 2 · 2 · 2 · 2, 36 \u003d 1 · 2 · 2 · 3 · 3.
2. Find some multipliers: 1, 2 and 2.
3. Their work will give Nod: 1 · 2 · 2 \u003d 4
4. We will find NOK now: To do this, I will find the NOK (12, 32): 12 · 32/4 \u003d 96.
5. To find the NOC of all three numbers, you need to find a node (96, 36): 96 \u003d 1 · 2 · 2 · 2 · 2 · 2 · 3, 36 \u003d 1 · 2 · 2 · 3 · 3, node \u003d 1 · 2 · 2 · 3 \u003d 12.
6. NOK (12, 32, 36) \u003d 96 · 36/12 \u003d 288.