This online calculator decomposes the numbers to the simple factors by the method of intersecting simple divisors. If the number is large, then for the convenience of the presentation, use the separator of discharges.
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Decomposition of the number on simple factors - theory, algorithm, examples and solutions
One of the simplest ways to decompose the number on simple factors is to check whether this number is divided by 2, 3, 5, ..., i.d., i.e. Check if the number is divided into a number of prime numbers. If the number n. It is not divided into any simple number before, then this number is simple, because If the number is composite, then there are at least two factor and both of them can no longer be.
Imagine the algorithm of the decomposition of the number n. on simple factors. Prepare in advance the table of prime numbers to s.\u003d. Denote a number of prime numbers through p. 1 , p. 2 , p. 3 , ...
Algorithm of decomposition of the number on simple dividers:
Example 1. Ensure the number 153 to simple multipliers.
Decision. We are enough to have a table of prime numbers to 
. 2, 3, 5, 7, 11.
Delim 153 at 2. 153 is not divided into 2 without a residue. Next, divide 153 to the next element of the table of prime numbers, i.e. at 3. 153: 3 \u003d 51. Fill the table:
Next, we check whether the number 17 to 3. The number 17 is not divided by 3. It is not divided into numbers 5, 7, 11. The next divider is more . Consequently 17 a simple number that is divided only to: 17: 17 \u003d 1. The procedure is stopped. Fill the table:
We choose those dividers on which the numbers 153, 51, 17 were divided without a balance, i.e. All numbers on the right side of the table. These are divisors 3, 3, 17. Now the number 153 can be represented as a product of prime numbers: 153 \u003d 3 · 3 · 17.
Example 2. Dispatch the number 137 to simple multipliers.
Decision. Calculate 
. So we need to check the divisibility of the number 137 to the simple numbers to 11: 2,3,5,7,11. Alternately dividing the number 137 to these numbers, we find out that the number 137 is not divided into one of the numbers 2,3,5,7,11. Therefore, 137 simple.
Each natural number, except for the unit, has two or more divisors. For example, the number 7 is divided without a residue for 1 and 7, that is, it has two divisors. And in the number 8, dividers 1, 2, 4, 8, that is, as much as 4 divisors at once.
What is the difference between simple and constituent numbers
Numbers that have more than two divisors are called composite. Numbers that have only two divisors: a unit and itself this number is called simple numbers.
The number 1 has only one to divide, namely, this is the number itself. The unit does not apply to any simple nor to the compound numbers.
- For example, the number 7 is simple, and the number 8 is composite.
The first 10 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Number 2 single even simple number, all other simple numbers are odd.
The number 78 is composite, since in addition to 1 and itself, it is also divided into 2. When dividing 2, we obtain 39. That is 78 \u003d 2 * 39. In such cases, they say that the number decomposed on multipliers 2 and 39.
Any composite number can be decomposed on two factors, each of which is more than 1. With a simple number, such a focus will not roll. So it goes.
Decomposition of a number of simple factors
As noted above, any composite number can be decomposed into two factors. Take, for example, the number 210. This number can be decomposed on two factor 21 and 10. But the numbers 21 and 10 are also composite, decompose them on two factors. We obtain 10 \u003d 2 * 5, 21 \u003d 3 * 7. And in the end, the number 210 decomposed already on 4 factor: 2,3,5,7. These numbers are already simple and cannot be decomposed. That is, we laid out the number 210 on simple multipliers.
In the decomposition of the components on simple multipliers, they are usually recorded in ascending order.
It should be remembered that any composite number can be decomposed on simple multipliers and with the only way, with an accuracy of rearrangement.
- Usually, with the decomposition of the number to simple multipliers, they use signs of divisibility.
Spread the number 378 for simple factors
We will record numbers separating them with a vertical feature. The number 378 is divided into 2, as it ends with 8. When divisions, we obtain the number 189. The amount of numbers of the number 189 is divided by 3, which means the number 189 is divided by 3. As a result, we obtain 63.
The number 63 is also divided into 3, on the basis of divisibility. We obtain 21, the number 21 can be divided into 3 again, we get 7. The seven is divided only on itself, we get a unit. This is finished division. To the right after the feature turned out to be simple multipliers, for which the number 378 is folded.
378|2
189|3
63|3
21|3
What does it mean to decompose on simple multipliers? How to do it? What can be found in the decomposition of the number on simple multipliers? Answers to these questions are illustrated by specific examples.
Definitions:
Simple called the number that has exactly two different divisors.
Compound refer to the number that has more than two divisors.
Eliminate the natural number of multipliers - it means to present it as a product of natural numbers.
Eliminate the natural number on simple multipliers - it means to present it as a product of prime numbers.
Remarks:
- In the decomposition of a simple number, one of the multipliers is equal to one, and the other is the most of this number.
- It makes sense to talk about the decomposition of units on multipliers.
- The composite number can be decomposed on multipliers, each of which is different from 1.
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Spread the number 150 for multipliers. For example, 150 is 15 multiplied by 10. 15 is a composite number. It can be decomposed on simple multipliers 5 and 3. 10 is a composite number. It can be decomposed on simple multipliers 5 and 2. After writing instead of 15 and 10 of their decomposition into simple multipliers, we received a decomposition of the number 150. |
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The number 150 can be distributed differently on multipliers. For example, 150 is a product of numbers 5 and 30. 5 - the number is simple. 30 is the number of composite. It can be represented as a piece of 10 and 3. 10 - the number of composite. It can be decomposed on simple multipliers 5 and 2. We obtained a decomposition of the number 150 on simple factors in another way. |
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Note that the first and second decomposition is the same. They differ only by the procedure for finding multipliers. It is customary to record multipliers in ascending order. |
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Any composite number can be decomposed on simple factors the only way up to the procedure for multipliers. |
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With the decomposition of large numbers to simple multipliers, they use the record in the column:
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The smallest simple number to which 216 is divided is 2. We divide 216 to 2. Receive 108. |
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The resulting number 108 is divided into 2. Perform division. We obtain as a result of 54. |
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According to a sign of divisibility for 2, 54 is divided into 2. By performing division, we get 27. |
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The number 27 ends on an odd digit 7. It It is not divided into 2. The next simple number is 3. We divide 27 to 3. Receive 9. The smallest simple The number on which is divided by 9 is 3. Three - itself is a simple number, it is divided into itself and per unit. We divide 3 on yourself. As a result, we got 1. |
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- The number is divided only to those simple numbers that are part of its decomposition.
- The number is divided only on those constituents whose decomposition on simple factors is completely contained.
Consider examples:
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4900 is divided into simple numbers 2, 5 and 7. (They are included in the expansion of the number 4900), but not divisible, for example, by 13. |
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11 550 75. This is so, because the decomposition of the number 75 is fully contained in the decomposition of the number 11550. As a result of division, there will be a product of multipliers 2, 7 and 11. 11550 is not divided into 4 because there are extra two in the expansion of four. |
Find a private from the division of the number A to the number B if these numbers are folded into simple factors as follows A \u003d 2 ∙ 2 ∙ 2 ∙ 3 \u200b\u200b∙ 3 ∙ 3 ∙ 5 ∙ 5 ∙ 19; B \u003d 2 ∙ 2 ∙ 3 \u200b\u200b∙ 3 ∙ 5 ∙ 19
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The decomposition of the number B is fully contained in the decomposition of the number a. |
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The result of division A on B is a product of the three numbers remaining in the expansion. So, the answer is: 30. |
Bibliography
- Vilekin N.Ya., Zhokhov V.I., Chesnokov A.S., Schwarzburg S.I. Mathematics 6. - M.: Mnemozina, 2012.
- Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics Grade 6. - Gymnasium. 2006.
- Depman I.Ya., Vilenkin N.Ya. Behind the pages of the textbook of mathematics. - M.: Enlightenment, 1989.
- Rurukin A.N., Tchaikovsky I.V. Tasks at the rate of mathematics 5-6 class. - M.: Zh MEPI, 2011.
- Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. Manual for students of the 6th grade of the correspondence school of MEPI. - M.: Zh MEPI, 2011.
- Chevrine L.N., Gain A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook - Interlocutor for 5-6 High School Classes. - M.: Enlightenment, Library of Mathematics Teacher, 1989.
- Internet portal Matematika-na.ru ().
- Internet portal Math-Portal.ru ().
Homework
- Vilekin N.Ya., Zhokhov V.I., Chesnokov A.S., Schwarzburg S.I. Mathematics 6. - M.: Mnemozina, 2012. No. 127, No. 129, No. 141.
- Other tasks: № 133, № 144.
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(except 0 and 1) have a minimum of two divisors: 1 and himself. Numbers that do not have other divisors are called simple numbers. Numbers having other dividers are called compound(or complicated) numbers. Simple numbers are an infinite set. Below are simple numbers that are not exceeding 200:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,
47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101,
103, 107, 109, 113, 127, 131, 137, 139, 149, 151,
157, 163, 167, 173, 179, 181, 191, 193, 197, 199.
Multiplication - One of the four major arithmetic action, a binary mathematical operation in which one argument develops as many times as the other shows. In arithmetic, the multiplication understands a brief entry of the addition of the specified number of the same components.
for example, Recording 5 * 3 denotes "fold three fives", that is, 5 + 5 + 5. The result of multiplication is called work , and multiplying numbers - multipliers or in fact. The first factor is sometimes called " multiplicand».
Any composite number can be decomposed on simple multipliers. With any method, one and the same decomposition is obtained if not considering the procedure for recording multipliers.
Decomposition of the number of multipliers (factorization).
Displays for multipliers (factorization) - Bust dividers - the algorithm for factorization or testing of the simplicity of the number by full of all possible potential divisors.
Those., Simple language, factorization is the name of the process of decomposition of numbers for multipliers, expressed by scientific language.
Sequence of actions with decomposition into simple factors:
1. Check if the proposed number is simple.
2. If not, then we select, guided by signs of division divisor, from the simple numbers since the smallest (2, 3, 5 ...).
3. We repeat this action until the private will be a simple number.