Remain we will find the nodes. Algorithm Euclida - Finding the greatest common divider

The greatest common divisor (Node) Two numbers called the largest number to which both numbers will be divided without a residue.

Designation: Node (a; c).

EXAMPLE. We find the nodes of numbers 4 and 6.

Number 4 without a residue is divided by: 1, 2 and 4.
Number 6 without a residue is divided by: 1, 2, 3 and 6.
The greatest common divider of numbers 4 and 6 will be the number 2.
Node (4; 6) \u003d 2

This is a simple example. And what about the large numbers for which it is necessary to find nodes?

In such cases, the numbers are declined to simple factors, after which the same multipliers in both expansions are noted - the product of marked simple multipliers and will be the NOD.

EXAMPLE. We will find the nodes of numbers 81 and 45.

81 = 3 · 3 · 3 · 3 45 \u003d 3 · 3 · 5 node (81; 45) \u003d 3 · 3 = 9

In cases where two numbers have no identical simple multipliers, the only natural number that will be divided into such numbers will be 1. NODs of such numbers \u003d 1. For example: node (7; 15) \u003d 1.

What is NOK.

The number A is called multiple The number in, if it is divided into without a residue (aimed). For example, 10 is divided by 5, therefore, 10 times 5; 11 is not divided by focus on 5, therefore, 11 is not multiple 5.

The smallest common paint (NOC) of two natural numbers is called the smallest number, multiple these two numbers.

Designation: NOK (A; B).

Nok finding rule:

decompose both numbers on simple factors, note the same simple multipliers in both decompositions, if any;
the product of all simple multipliers of one of the numbers (actually, the number of) and all non-marked multipliers will make NOC.

EXAMPLE. We find NOC numbers 81 and 45.

81 = 3 · 3 · 3 · 3 45 \u003d 3 · 3 · 5 NOC (81; 45) \u003d 81 · 5 \u003d 405

405 is the smallest multiple for numbers 81 and 45: 405/81 \u003d 5; 405/45 \u003d 9.

If two numbers have no identical simple multipliers, then the NOC for such numbers will be equal to the product of these numbers.

14 \u003d 2 · 7 15 \u003d 3 · 5 NOC (14; 15) \u003d 14 · 15 \u003d 210

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.

The greatest common divisel

Definition 2.

If the natural number A is divided into the natural number of $ b $, then $ b $ is called the number divider of $ A $, and the number $ a $ is called a multiple of the $ b $.

Let $ a $ and $ b $ -nutral numbers. The number $ C $ is called a common divider and for $ a $ and for $ b $.

Many common dividers of $ a $ and $ b $ are of course, since none of these divisors can be more than $ a $. It means that there are the largest among these divisors, which is called the greatest common divider of $ a $ and $ b $ and write records for its designation:

$ Node \\ (a; b) \\ or \\ d \\ (a; b) $

To find the largest common divider of two, numbers need:

Find a product of numbers found in step 2. The resulting number will be the desired largest common divisor.

Example 1.

Find Nodes $ 121 $ and $ 132. $

$ 242 \u003d 2 \\ CDOT 11 \\ CDOT $ 11

$ 132 \u003d 2 \\ CDOT 2 \\ CDOT 3 \\ CDOT 11 $

Select numbers that are included in the decomposition of these numbers

$ 242 \u003d 2 \\ CDOT 11 \\ CDOT $ 11

$ 132 \u003d 2 \\ CDOT 2 \\ CDOT 3 \\ CDOT 11 $

Find the product of the numbers found in step 2. The number has been received and will be the famous largest common divisor.

$ Node \u003d 2 \\ cdot 11 \u003d 22 $

Example 2.

Find a node of homorals $ 63 $ and $ 81 $.

We will find according to the presented algorithm. For this:

Spreads the numbers on simple multipliers

$ 63 \u003d 3 \\ CDOT 3 \\ CDOT $ 7

$ 81 \u003d 3 \\ CDOT 3 \\ CDOT 3 \\ CDOT $ 3

Choose the numbers that are included in the decomposition of these numbers

$ 63 \u003d 3 \\ CDOT 3 \\ CDOT $ 7

$ 81 \u003d 3 \\ CDOT 3 \\ CDOT 3 \\ CDOT $ 3

We will find a product of the numbers found in step 2. The received number and will be the desired largest common divisor.

$ Node \u003d 3 \\ cdot 3 \u003d 9 $

It is possible to find a node of two numbers in a different way, using many numbers dividers.

Example 3.

Find a node number $ 48 and $ 60 $.

Decision:

We find many divisors of the number $ 48 $: $ \\ left \\ ((\\ rm 1,2,3.4.6,8,12,16,24,48) \\ right \\) $

Now we find many divisors of the number $ 60 $: $ \\ \\ left \\ ((\\ rm 1,2,3,4,5,6,10,12,15,20,30,60) \\ Right \\) $

We will 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 $ 48 and $ 60 $. The largest element in this set will be the number of $ 12 $. So the greatest common divider of $ 48 $ and $ 60 $ will be $ 12 $.

Nok.

Definition 3.

Common multiple natural numbers $ a $ and $ b $ is called a natural number that is multiple and $ a $ and $ b $.

Common multiple numbers are called numbers that are divided into source without a residue. For example, forms $ 25 and $ 50 $ 50 by common multiple numbers $ 50,100,150,200 $, etc

The smallest of the total multiple will be called the smallest common multiple and is denoted by the NOC $ (a; b) $ or k $ (a; b). $

To find the NOC of two numbers, you need:

Dispatch numbers for simple factors
To write down the multipliers in the first number and add multipliers to them, which are part of the second and do not go to the first

Example 4.

Finding NOC numbers $ 99 $ and $ 77 $.

We will find according to the presented algorithm. For this

Dispatch numbers for simple factors

$ 99 \u003d 3 \\ CDOT 3 \\ CDOT $ 11

To write down the multipliers in the first

add multipliers to them, which are part of the second and do not go to the first

Find a product of numbers found in step 2. The number has been received and will be the desired smallest common

$ Nok \u003d 3 \\ CDOT 3 \\ CDOT 11 \\ CDOT 7 \u003d $ 693

Drawing up lists of dividers of numbers is often very laborious occupation. There is a way to find a node called the Euclidea algorithm.

The statements on which the Euclid algorithm is founded:

If $ a $ and $ b $ - represents, and $ a \\ vdots b $, then $ D (a; b) \u003d b $

If $ a $ and $ b $ - represents, such that $ b

Using $ D (a; b) \u003d D (A-B; b) $, one can consistently reduce the numbers under consideration until we do to such a pair of numbers that one of them is divided into another. Then the smaller of these numbers will be the desired largest common divider for numbers $ a $ and $ b $.

Properties Nod and Nok

Any common multiple numbers $ A $ and $ B $ is divided into k $ (a; b) $
If $ a \\ vdots b $, then to $ (a; b) \u003d a $

If to $ (a; b) \u003d k $ and $ m $ -natural number, then to $ (am; bm) \u003d km $

If $ d $--paper divider for $ a $ and $ b $, then to ($ \\ FRAC (A) (D); \\ FRAC (B) (D) $) \u003d $ \\ \\ FRAC (K) (D) $

If $ a \\ vdots C $ and $ b \\ vdots C $, then $ \\ FRAC (AB) (C) $ - total multiple numbers $ a $ and $ b $

For any natural numbers $ a $ and $ b $ equality is performed

$ D (a; b) \\ cdot to (a; b) \u003d AB $

Any common divider of numbers $ a $ and $ b $ is a divider of the number $ d (a; b) $

The greatest common divisel

Definition 2.

If the natural number A is divided into the natural number of $ b $, then $ b $ is called the number divider of $ A $, and the number $ a $ is called a multiple of the $ b $.

Let $ a $ and $ b $ -nutral numbers. The number $ C $ is called a common divider and for $ a $ and for $ b $.

$ Node \\ (a; b) \\ or \\ d \\ (a; b) $

To find the largest common divider of two, numbers need:

Find a product of numbers found in step 2. The resulting number will be the desired largest common divisor.

Example 1.

Find Nodes $ 121 $ and $ 132. $

$ 242 \u003d 2 \\ CDOT 11 \\ CDOT $ 11

$ 132 \u003d 2 \\ CDOT 2 \\ CDOT 3 \\ CDOT 11 $

Select numbers that are included in the decomposition of these numbers

$ 242 \u003d 2 \\ CDOT 11 \\ CDOT $ 11

$ 132 \u003d 2 \\ CDOT 2 \\ CDOT 3 \\ CDOT 11 $

Find the product of the numbers found in step 2. The number has been received and will be the famous largest common divisor.

$ Node \u003d 2 \\ cdot 11 \u003d 22 $

Example 2.

Find a node of homorals $ 63 $ and $ 81 $.

We will find according to the presented algorithm. For this:

Spreads the numbers on simple multipliers

$ 63 \u003d 3 \\ CDOT 3 \\ CDOT $ 7

$ 81 \u003d 3 \\ CDOT 3 \\ CDOT 3 \\ CDOT $ 3

Choose the numbers that are included in the decomposition of these numbers

$ 63 \u003d 3 \\ CDOT 3 \\ CDOT $ 7

$ 81 \u003d 3 \\ CDOT 3 \\ CDOT 3 \\ CDOT $ 3

We will find a product of the numbers found in step 2. The received number and will be the desired largest common divisor.

$ Node \u003d 3 \\ cdot 3 \u003d 9 $

It is possible to find a node of two numbers in a different way, using many numbers dividers.

Example 3.

Find a node number $ 48 and $ 60 $.

Decision:

We find many divisors of the number $ 48 $: $ \\ left \\ ((\\ rm 1,2,3.4.6,8,12,16,24,48) \\ right \\) $

Now we find many divisors of the number $ 60 $: $ \\ \\ left \\ ((\\ rm 1,2,3,4,5,6,10,12,15,20,30,60) \\ Right \\) $

Nok.

Definition 3.

Common multiple natural numbers $ a $ and $ b $ is called a natural number that is multiple and $ a $ and $ b $.

Common multiple numbers are called numbers that are divided into source without a residue. For example, forms $ 25 and $ 50 $ 50 by common multiple numbers $ 50,100,150,200 $, etc

The smallest of the total multiple will be called the smallest common multiple and is denoted by the NOC $ (a; b) $ or k $ (a; b). $

To find the NOC of two numbers, you need:

Dispatch numbers for simple factors
To write down the multipliers in the first number and add multipliers to them, which are part of the second and do not go to the first

Example 4.

Finding NOC numbers $ 99 $ and $ 77 $.

We will find according to the presented algorithm. For this

Dispatch numbers for simple factors

$ 99 \u003d 3 \\ CDOT 3 \\ CDOT $ 11

To write down the multipliers in the first

add multipliers to them, which are part of the second and do not go to the first

Find a product of numbers found in step 2. The number has been received and will be the desired smallest common

$ Nok \u003d 3 \\ CDOT 3 \\ CDOT 11 \\ CDOT 7 \u003d $ 693

Drawing up lists of dividers of numbers is often very laborious occupation. There is a way to find a node called the Euclidea algorithm.

The statements on which the Euclid algorithm is founded:

If $ a $ and $ b $ - represents, and $ a \\ vdots b $, then $ D (a; b) \u003d b $

If $ a $ and $ b $ - represents, such that $ b

Properties Nod and Nok

Any common multiple numbers $ A $ and $ B $ is divided into k $ (a; b) $
If $ a \\ vdots b $, then to $ (a; b) \u003d a $

If to $ (a; b) \u003d k $ and $ m $ -natural number, then to $ (am; bm) \u003d km $

If $ d $--paper divider for $ a $ and $ b $, then to ($ \\ FRAC (A) (D); \\ FRAC (B) (D) $) \u003d $ \\ \\ FRAC (K) (D) $

If $ a \\ vdots C $ and $ b \\ vdots C $, then $ \\ FRAC (AB) (C) $ - total multiple numbers $ a $ and $ b $

For any natural numbers $ a $ and $ b $ equality is performed

$ D (a; b) \\ cdot to (a; b) \u003d AB $

Any common divider of numbers $ a $ and $ b $ is a divider of the number $ d (a; b) $

To find the smallest common pain (NOC) and the greatest common divisel (Nod) Two numbers Use our online calculator:

Enter numbers: and
Nok:
Node:

Determine

Just enter numbers and get the result.

How to find a nok two numbers

The smallest total multiple (NOK) Two or more numbers are the smallest number that can be divided into each of these numbers without a residue.

In order to find the smallest total multiple (NOC) two numbers, you can use the following algorithm (grade 5):

Both numbers (first the highest number).
Compare multiple numbers with fewer multipliers. We highlight all multipliers of a smaller number that are not more.
Add the dedicated multipliers of a smaller number to multiple faults.
I will find the NOC, moving a number of multipliers received in paragraph 3.

Example

For example, we define the NOC numbers 8 and 22 .

1) Unlock on simple factors:

2) Allocate all multipliers of 8, which are not 22:

8 = 2⋅2 ⋅2

3) Add the selected multipliers of 8 to multipliers of the 22nd:

NOK (8; 22) \u003d 2 · 11 · 2 · 2

4) Calculate the NOC:

NOK (8; 22) \u003d 2 · 11 · 2 · 2 \u003d 88

How to find a node two numbers

The greatest common divider (node) Two or more numbers is the greatest natural integer on which these numbers can be divided without a residue.

To find the largest common divider (node) of the two numbers, first need to decompose them on simple multipliers. Then you need to allocate general factors that are also available at the first number and the second. Moving them - it will be the node. To better understand the algorithm, consider an example:

Example

For example, we define the nodes 20 and 30 .

20 = 2 ⋅2⋅5

30 = 2 ⋅3⋅5

Node (20.30) = 2⋅5 = 10