How to find the smallest common multiple, NOC for two or more numbers. The smallest total multiple (NOC): definition, examples and properties

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:

Fold, multiply, divide, lead to common denominator And other arithmetic action is 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.

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

Most simple way Calculations of the greatest general divider 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.

Consider three ways to find the smallest common multiple.

Laying by expansion on multipliers

The first method is to find the smallest common multiple by decomposition of these numbers on simple factors.

Suppose we need to find NOC numbers: 99, 30 and 28. For this, we will decompose each of these numbers to simple multipliers:

To share the desired number 99, by 30 and 28, it is necessary and enough for all the simple factors of these divisors to be included in it. To do this, we need to take all the simple factors of these numbers to the greatest extent and multiply them with each other:

2 2 · 3 2 · 5 · 7 · 11 \u003d 13 860

Thus, the NOK (99, 30, 28) \u003d 13 860. No other number is less than 13,860 by 99, by 30 and by 28.

To find the smallest common multiple data of numbers, you need to decompose them on simple multipliers, then take every simple multiplier with the greatest indicator of the degree, with which it is found, and multiply these multipliers with each other.

Since mutually simple numbers do not have common simple multipliers, their smallest common multiple is equal to the product of these numbers. For example, three numbers: 20, 49 and 33 are mutually simple. therefore

NOC (20, 49, 33) \u003d 20 · 49 · 33 \u003d 32 340.

In the same way, it is necessary to act when the smallest common multiple of various simple numbers is found. For example, NOK (3, 7, 11) \u003d 3 · 7 · 11 \u003d 231.

Finding the selection

The second method is to find the smallest common multiple by the selection.

Example 1. When the largest of these numbers is divided into other data of the number, the NOC of these numbers is equal to greater of them. For example, four numbers are given: 60, 30, 10 and 6. Each of them is divided by 60, therefore:

NOK (60, 30, 10, 6) \u003d 60

In other cases, the following procedure is used to find the smallest total:

1. Determine the largest number from these numbers.
2. Next, we find numbers, multiple the greatest number, multiplying it to integers In order of their increase and checking whether the rest of the number of numbers obtained is divided into the result.

Example 2. Three numbers 24, 3 and 18 are given. We determine the largest of them - this is the number 24. Next, we find numbers of multiples 24, checking if each of them is divided by 18 and 3:

24 · 1 \u003d 24 - divided by 3, but not divided by 18.

24 · 2 \u003d 48 - divided by 3, but not divided by 18.

24 · 3 \u003d 72 - divided by 3 and 18.

Thus, the NOC (24, 3, 18) \u003d 72.

Finding a consistent NOC

The third way is to find the smallest common pain in the sequential finding of the NOC.

The NOC of the two data data is equal to the product of these numbers divided into their largest common divisor.

Example 1. Find the NOC of the two data data: 12 and 8. We define their largest common divisor: node (12, 8) \u003d 4. Reduce the number of numbers:

We divide the work on their nodes:

Thus, the NOK (12, 8) \u003d 24.

To find the Nok three or more numbers, the following procedure is used:

1. First find the NOC some of the two numbers.
2. Then, the NOC found the least common multiple and the third one.
3. Then, NOC obtained the smallest total multiple and fourth number, etc.
4. Thus, the search for NOC continues until there are numbers.

Example 2. Find the NOC of three data numbers: 12, 8 and 9. NOC numbers 12 and 8 We have already found in the previous example (this is the number 24). It remains to find the smallest total multiple number 24 and the third of this number - 9. We define their largest common divisor: nodes (24, 9) \u003d 3. Reduce the NOC with number 9:

We divide the work on their nodes:

Thus, the NOC (12, 8, 9) \u003d 72.

A multiple number is a number that is divided into a given number without a residue. The smallest common multiple (NOC) groups of numbers is the smallest number that is divided without a residue for each number of the group. To find the smallest common multiple, you need to find simple multipliers of these numbers. NOCs can also be calculated using a number of other methods that are applicable to groups of two or more numbers.

Steps

A number of multiple numbers

Look at the data of the number. The method described here is better to apply when two numbers are given, each of which is less than 10. If large numbers are given, use the other method.

• For example, find the smallest common multiple numbers 5 and 8. These are small numbers, so this method can be used.

1. A multiple number is a number that is divided into a given number without a residue. Multiple numbers can be viewed in the multiplication table ..

   • For example, the numbers that are multiple 5 are: 5, 10, 15, 20, 25, 30, 35, 40.

2. Write down a number of numbers that are multiple the first number. Do it under multiple numbers of the first number to compare two rows of numbers.

   • For example, the numbers that are multiple 8 are: 8, 16, 24, 32, 40, 48, 56, and 64.

3. Find the smallest number that is present in both rows of multiple numbers. You may have to write long rows of multiple numbers to find the total number. The smallest number that is present in both rows of multiple numbers is the smallest common.

   • For example, the smallest number that is present in the rows of multiple numbers 5 and 8 is the number 40. Therefore, 40 is the smallest total multiple numbers 5 and 8.

   Decomposition of simple factors

   1. Look at the data of the number. The method described here is better to apply when two numbers are given, each of which is more than 10. If smaller numbers are given, use the other method.

      • For example, find the smallest general multiple numbers 20 and 84. Each of the numbers is greater than 10, so this method can be used.

   2. Spread the first number to simple factors. That is, you need to find such simple numbers, when multiplying which this number will turn out. Finding simple multipliers, write them down in the form of equality.

      • For example, 2 × 10 \u003d 20 (\\ displayStyle (\\ MathBF (2)) \\ Times 10 \u003d 20) and 2 × 5 \u003d 10 (\\ DisplayStyle (\\ MathBF (2)) \\ Times (\\ MathBF (5)) \u003d 10). Thus, simple multipliers of the number 20 are numbers 2, 2 and 5. Record them as an expression :.

   3. Spread the second number on simple factors. Do it the same way as you laid out the first number to multipliers, that is, find such simple numbers, with multiplies this number.

      • For example, 2 × 42 \u003d 84 (\\ DISPLAYSTYLE (\\ MathBF (2)) \\ Times 42 \u003d 84), 7 × 6 \u003d 42 (\\ displayStyle (\\ Mathbf (7)) \\ Times 6 \u003d 42) and 3 × 2 \u003d 6 (\\ DisplayStyle (\\ MathBF (3)) \\ Times (\\ MathBF (2)) \u003d 6). Thus, the simple multipliers of the number 84 are numbers 2, 7, 3 and 2. Record them as an expression :.

   4. Write down the multipliers common to both numbers. Write down such multipliers in the form of multiplication operation. As each multiplier records, jump it in both expressions (expressions that describe the decomposition of numbers to simple multipliers).

      • For example, common for both numbers is multiplier 2, so write 2 × (\\ DisplayStyle 2 \\ Times) And cross out 2 in both expressions.
      • Common for both numbers is another multiplier 2, so write 2 × 2 (\\ DisplayStyle 2 \\ Times 2) And cross out the second 2 in both expressions.

   5. Add the remaining multipliers to the multiplication operation. These are multipliers that are not crossed in both expressions, that is, faults that are not common to both numbers.

      • For example, in expression 20 \u003d 2 × 2 × 5 (\\ DISPLAYSTYLE 20 \u003d 2 \\ Times 2 \\ Times 5) Crushed both twos (2), because they are common factors. Multiplier 5 will not cross out, therefore, the multiplication is recorded as follows: 2 × 2 × 5 (\\ DisplayStyle 2 \\ Times 2 \\ Times 5)
      • In expression 84 \u003d 2 × 7 × 3 × 2 (\\ DISPLAYSTYLE 84 \u003d 2 \\ TIMES 7 \\ TIMES 3 \\ TIMES 2) Also crossed out both twins (2). The multipliers 7 and 3 are not crossed out, so the multiplication operation is recorded: 2 × 2 × 5 × 7 × 3 (\\ DisplayStyle 2 \\ Times 2 \\ Times 5 \\ Times 7 \\ Times 3).

   6. Calculate the smallest common multiple. To do this, multiply the numbers in the recorded multiplication operation.

      • For example, 2 × 2 × 5 × 7 × 3 \u003d 420 (\\ DisplayStyle 2 \\ Times 2 \\ Times 5 \\ Times 7 \\ Times 3 \u003d 420). Thus, the smallest overall multiple 20 and 84 is 420.

   Finding common divisors

   1. Draw the grid as to play in noliki cross. Such a mesh is two parallel straight lines, which intersect (at right angles) with other two parallel straight. Thus, there are three lines and three columns (the grid is very similar to the # icon). Write the first number in the first row and the second column. Write the second number in the first line and the third column.

      • For example, find the smallest overall multiple numbers 18 and 30. Number 18 Write in the first line and the second column, and write the number 30 in the first line and the third column.

   2. Find a divider common to both numbers. Write it down in the first line and first column. It is better to look for simple dividers, but this is not a prerequisite.

      • For example, 18 and 30 is even numbers, Therefore, their common divider will be a number 2. Thus, write 2 in the first row and the first column.

   3. Divide each number on the first divider. Each privately recorded under the appropriate number. Private is the result of dividing two numbers.

      • For example, 18 ÷ 2 \u003d 9 (\\ DISPLAYSTYLE 18 \\ DIV 2 \u003d 9), Therefore, write 9 under 18.
      • 30 ÷ 2 \u003d 15 (\\ DisplayStyle 30 \\ DIV 2 \u003d 15), Therefore, write 15 under 30.

   4. Find a divider common to both private. If there is no such divider, skip the following two steps. Otherwise, the divider will write down in the second line and first column.

      • For example, 9 and 15 are divided into 3, so write 3 in the second row and the first column.

   5. Divide each private on the second divider. Each division result is recorded under the appropriate private.

      • For example, 9 ÷ 3 \u003d 3 (\\ DisplayStyle 9 \\ Div 3 \u003d 3), Therefore, write 3 under 9.
      • 15 ÷ 3 \u003d 5 (\\ DISPLAYSTYLE 15 \\ DIV 3 \u003d 5), Therefore, write 5 under 15.

   6. If necessary, add the grid with additional cells. Repeat the described actions until the private will not have a common divider.

   7. Circle numbers in the first column and the last row of the grid. Then the selected numbers record as multiplication operation.

      • For example, numbers 2 and 3 are in the first column, and the numbers 3 and 5 are in the last line, so the multiplication operation is recorded as follows: 2 × 3 × 3 × 5 (\\ DisplayStyle 2 \\ Times 3 \\ Times 3 \\ Times 5).

   8. Find the result of multiplication of numbers. So you will calculate the smallest general multiple of two numbers data.

      • For example, 2 × 3 × 3 × 5 \u003d 90 (\\ DISPLAYSTYLE 2 \\ TIMES 3 \\ TIMES 3 \\ TIMES 5 \u003d 90). Thus, the smallest total multiple 18 and 30 is 90.

   Algorithm Euclida

   1. Remember the terminology associated with the division operation. Delimi is the number that is divided. The divider is the number for which they divide. Private is the result of dividing two numbers. The residue is the number remaining when dividing two numbers.

      • For example, in expression 15 ÷ 6 \u003d 2 (\\ DisplayStyle 15 \\ Div 6 \u003d 2) Ost. 3:
        15 - this is divisible
        6 is a divider
        2 is a private
        3 is the residue.

We will proceed to the study of the smallest common multiple two or more numbers. In the section, we will give the Definition of the term, consider the theorem that establishes the link between the smallest common multiple and the largest common divisor, we give examples of solving problems.

Common multiples - definition, examples

In this topic, we will only be interested in the total multiple integers other than zero.

Definition 1.

Total multiple integers - This is such an integer that is multiple of all these numbers. In fact, this is any integer that can be divided into any of these numbers.

The determination of common multiple numbers relates to two, three and more integer numbers.

Example 1.

According to the above definition for the number 12 by community multiple numbers will be 3 and 2. Also, the number 12 will be a common multiple for numbers 2, 3 and 4. Numbers 12 and - 12 are common multiple numbers for numbers ± 1, ± 2, ± 3, ± 4, ± 6, ± 12.

At the same time, the total multiple number for numbers 2 and 3 will be numbers 12, 6, - 24, 72, 468, - 100 010 004 and a number of any other.

If we take the numbers that are divided into the first number from the pair and are not divided into second, then such numbers will not be general multiple. So, for numbers 2 and 3 numbers 16, - 27, 5 009, 27 001 will not be general multiple.

0 is a common multiple for any set of integers other than zero.

If you remember the property of divisibility regarding opposite numbersIt turns out that some integer k will be a common multiple data of numbers as well as the number - k. This means that common divisters can be both positive and negative.

Is it possible to find NOC for all the numbers?

Common multiple can be found for any integers.

Example 2.

Suppose we are given K. integers A 1, A 2, ..., A K. The number we get during the multiplication of numbers a 1 · a 2 · ... · a k According to the property of divisibility, it will be divided into each of the multipliers, which was included in the initial work. This means that the number of numbers A 1, A 2, ..., A Kit is the smallest common to these numbers.

How many common multiple data can have data integers?

A group of integers may have a large number of common multiples. In fact, their number is infinite.

Example 3.

Suppose we have some number k. Then the product of the numbers k · z, where Z is an integer, will be a common multiple numbers K and Z. Taking into account the fact that the number of numbers is infinite, the number of common multiple is infinite.

The smallest total multiple (NOC) - definition, designation and examples

Recall the concept of the smallest number from this set of numbers we were viewed in the "Comparison of integers" section. Taking into account this concept, we formulate the definition of the smallest overall multiple, which has among all common multiples the greatest practical significance.

Definition 2.

The smallest total multiple data of integers - This is the smallest positive common multiple of these numbers.

The smallest overall multiple exists for any number of data data. The most used to designate the concept in the reference book is the abbreviation of NOC. A brief record of the smallest total multiple for numbers A 1, A 2, ..., A K will have kind of nok (A 1, A 2, ..., A K).

Example 4.

The smallest general multiple numbers 6 and 7 is 42. Those. NOK (6, 7) \u003d 42. The smallest total multiple of four numbers - 2, 12, 15 and 3 will be 60. A brief entry will be viewed NOC (- 2, 12, 15, 3) \u003d 60.

Not for all groups of these numbers, the smallest common is clear. Often it has to be calculated.

Communication between NOC and NOD

The smallest total multiple and the largest common divisor is interconnected. The relationship between concepts establishes the theorem.

Theorem 1.

The smallest general multiple of two positive integers A and B is equal to the product of 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).

Proof 1.

Suppose we have a number M, which is multiple of numbers a and b. If the number M is divided into A, there is also some integer Z , at which equality is right M \u003d a · k. According to the definition of divisibility, if M is divided into B., so then A · K. divided by B..

If we enter a new designation for NOD (A, B) as D., we can use equality a \u003d a 1 · d and b \u003d b 1 · d. At the same time, both equalities will be mutually simple numbers.

We have already set up above that A · K. divided by B.. Now this condition can be written as follows:
a 1 · d · k divided by B 1 · Dthat is equivalent to the condition A 1 · K divided by B 1. According to the properties of divisibility.

According to the property of mutually simple numbers, if A 1. and B 1. - Mutually simple numbers, A 1. Not divided by B 1. despite the fact that A 1 · K divided by B 1.T. B 1. must be shared K..

In this case it will appropriate to assume that there is a number T., for which k \u003d b 1 · t, and since B 1 \u003d B: DT. k \u003d b: d · t.

Now instead k. Substitute in equality M \u003d a · k Expression of type B: D · T. This allows us to come to equality. M \u003d A · B: D · T. For T \u003d 1. We can get the smallest positive common multiple numbers a and b , equal A · B: D, provided that the numbers a and b positive.

So we proved that the NOK (A, B) \u003d A · B: Nod (A, B).

The establishment of a connection between NOC and NOD allows you to find the smallest common multiple through the largest common divisor of two and more data data.

Definition 3.

Theorem has two important consequences:

• the multiple of the smallest total multiple two numbers coincides with the common multiple of these two numbers;
• the smallest common multiple of mutually simple positive numbers A and B are equal to their work.

Justify these two facts is not difficult. Any common multiple M numbers A and B is determined by the equality M \u003d NOC (A, B) · T with some whole value t. Since a and b are mutually simple, then node (a, b) \u003d 1, therefore, NOK (A, B) \u003d A · B: NOD (A, B) \u003d A · B: 1 \u003d A · b.

The smallest total multiple of three and more numbers

In order to find the smallest general multiple of several numbers, it is necessary to consistently find the NOC of two numbers.

Theorem 2.

Let's pretend that A 1, A 2, ..., A K - These are some whole positive numbers. In order to calculate the NOK m K. these numbers, we need to consistently calculate m 2 \u003d nok (A 1, A 2), M 3 \u003d Nok. (m 2, a 3), ..., m k \u003d Nok. (m k - 1, a k).

Proof 2.

Proving the loyalty of the second theorem will help us the first consequence of the first theorem discussed in this topic. The arguments are built according to the following algorithm:

• common multiple numbers A 1. and A 2. coincide with multiple of their NOK, in fact, they coincide with multiple numbers m 2.;
• common multiple numbers A 1., A 2. and A 3. m 2. and A 3. M 3.;
• common multiple numbers A 1, A 2, ..., A K coincide with common multiple numbers M K - 1 and A K., therefore, coincide with multiple numbers M K.;
• due to the fact that the smallest positive multiple number M K. is the number of one M K.then the smallest common multiple numbers A 1, A 2, ..., A K is an M K..

So we proved the theorem.

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