Factoring powers of a number into prime factors. How to factor a number into a product of prime factors
This article gives answers to the question of factoring a number on a sheet. Let's look at the general idea of decomposition with examples. Let us analyze the canonical form of the expansion and its algorithm. All alternative methods will be considered using divisibility signs and multiplication tables.
What does it mean to factor a number into prime factors?
Let's look at the concept prime factors. It is known that every prime factor is a prime number. In a product of the form 2 · 7 · 7 · 23 we have that we have 4 prime factors in the form 2, 7, 7, 23.
Factorization involves its representation in the form of products of primes. If we need to decompose the number 30, then we get 2, 3, 5. The entry will take the form 30 = 2 · 3 · 5. It is possible that the multipliers may be repeated. A number like 144 has 144 = 2 2 2 2 3 3.
Not all numbers are prone to decay. Numbers that are greater than 1 and are integers can be factored. Prime numbers, when factored, are only divisible by 1 and themselves, so it is impossible to represent these numbers as a product.
When z refers to integers, it is represented as a product of a and b, where z is divided by a and b. Composite numbers are factored using the fundamental theorem of arithmetic. If the number is greater than 1, then its factorization p 1, p 2, ..., p n takes the form a = p 1 , p 2 , … , p n . The decomposition is assumed to be in a single variant.
Canonical factorization of a number into prime factors
During expansion, factors can be repeated. They are written compactly using degrees. If, when decomposing the number a, we have a factor p 1, which occurs s 1 times and so on p n – s n times. Thus the expansion will take the form a=p 1 s 1 · a = p 1 s 1 · p 2 s 2 · … · p n s n. This entry is called the canonical factorization of a number into prime factors.
When expanding the number 609840, we get that 609 840 = 2 2 2 2 3 3 5 7 11 11, its canonical form will be 609 840 = 2 4 3 2 5 7 11 2. Using canonical expansion, you can find all the divisors of a number and their number.
To correctly factorize, you need to have an understanding of prime and composite numbers. The point is to obtain a sequential number of divisors of the form p 1, p 2, ..., p n numbers a , a 1 , a 2 , … , a n - 1, this makes it possible to get a = p 1 a 1, where a 1 = a: p 1 , a = p 1 · a 1 = p 1 · p 2 · a 2 , where a 2 = a 1: p 2 , … , a = p 1 · p 2 · … · p n · a n , where a n = a n - 1: p n. Upon receipt a n = 1, then equality a = p 1 p 2 … p n we obtain the required decomposition of the number a into prime factors. notice, that p 1 ≤ p 2 ≤ p 3 ≤ … ≤ p n.
To find least common factors, you need to use a table of prime numbers. This is done using the example of finding the smallest prime divisor of the number z. When taking prime numbers 2, 3, 5, 11 and so on, and dividing the number z by them. Since z is not prime number, it should be taken into account that the smallest prime divisor will not be greater than z. It can be seen that there are no divisors of z, then it is clear that z is a prime number.
Example 1
Let's look at the example of the number 87. When it is divided by 2, we have that 87: 2 = 43 with a remainder of 1. It follows that 2 cannot be a divisor; division must be done entirely. When divided by 3, we get that 87: 3 = 29. Hence the conclusion is that 3 is the smallest prime divisor of the number 87.
When factoring into prime factors, you must use a table of prime numbers, where a. When factoring 95, you should use about 10 primes, and when factoring 846653, about 1000.
Let's consider the decomposition algorithm into prime factors:
- finding the smallest factor of divisor p 1 of a number a by the formula a 1 = a: p 1, when a 1 = 1, then a is a prime number and is included in the factorization, when not equal to 1, then a = p 1 · a 1 and follow to the point below;
- finding the prime divisor p 2 of a number a 1 by sequentially enumerating prime numbers using a 2 = a 1: p 2 , when a 2 = 1 , then the expansion will take the form a = p 1 p 2 , when a 2 = 1, then a = p 1 p 2 a 2 , and we move on to the next step;
- searching through prime numbers and finding a prime divisor p 3 numbers a 2 according to the formula a 3 = a 2: p 3 when a 3 = 1 , then we get that a = p 1 p 2 p 3 , when not equal to 1, then a = p 1 p 2 p 3 a 3 and move on to the next step;
- the prime divisor is found p n numbers a n - 1 by enumerating prime numbers with pn - 1, and a n = a n - 1: p n, where a n = 1, the step is final, as a result we get that a = p 1 · p 2 · … · p n .
The result of the algorithm is written in the form of a table with decomposed factors with a vertical bar sequentially in a column. Consider the figure below.
The resulting algorithm can be applied by decomposing numbers into prime factors.
When factoring into prime factors, the basic algorithm should be followed.
Example 2
Factor the number 78 into prime factors.
Solution
In order to find the smallest prime divisor, you need to go through all the prime numbers in 78. That is 78: 2 = 39. Division without a remainder means this is the first simple divisor, which we denote as p 1. We get that a 1 = a: p 1 = 78: 2 = 39. We arrived at an equality of the form a = p 1 · a 1 , where 78 = 2 39. Then a 1 = 39, that is, we should move on to the next step.
Let's focus on finding the prime divisor p2 numbers a 1 = 39. You should go through the prime numbers, that is, 39: 2 = 19 (remaining 1). Since division with a remainder, 2 is not a divisor. When choosing the number 3, we get that 39: 3 = 13. This means that p 2 = 3 is the smallest prime divisor of 39 by a 2 = a 1: p 2 = 39: 3 = 13. We obtain an equality of the form a = p 1 p 2 a 2 in the form 78 = 2 3 13. We have that a 2 = 13 is not equal to 1, then we should move on.
The smallest prime divisor of the number a 2 = 13 is found by searching through numbers, starting with 3. We get that 13: 3 = 4 (remaining 1). From this we can see that 13 is not divisible by 5, 7, 11, because 13: 5 = 2 (rest. 3), 13: 7 = 1 (rest. 6) and 13: 11 = 1 (rest. 2). It can be seen that 13 is a prime number. According to the formula it looks like this: a 3 = a 2: p 3 = 13: 13 = 1. We found that a 3 = 1, which means the completion of the algorithm. Now the factors are written as 78 = 2 · 3 · 13 (a = p 1 · p 2 · p 3) .
Answer: 78 = 2 3 13.
Example 3
Factor the number 83,006 into prime factors.
Solution
The first step involves factoring p 1 = 2 And a 1 = a: p 1 = 83,006: 2 = 41,503, where 83,006 = 2 · 41,503.
The second step assumes that 2, 3 and 5 are not prime divisors for the number a 1 = 41,503, but 7 is a prime divisor, because 41,503: 7 = 5,929. We get that p 2 = 7, a 2 = a 1: p 2 = 41,503: 7 = 5,929. Obviously, 83,006 = 2 7 5 929.
Finding the smallest prime divisor of p 4 to the number a 3 = 847 is 7. It can be seen that a 4 = a 3: p 4 = 847: 7 = 121, so 83 006 = 2 7 7 7 121.
To find the prime divisor of the number a 4 = 121, we use the number 11, that is, p 5 = 11. Then we get an expression of the form a 5 = a 4: p 5 = 121: 11 = 11, and 83,006 = 2 7 7 7 11 11.
For number a 5 = 11 number p 6 = 11 is the smallest prime divisor. Hence a 6 = a 5: p 6 = 11: 11 = 1. Then a 6 = 1. This indicates the completion of the algorithm. The factors will be written as 83 006 = 2 · 7 · 7 · 7 · 11 · 11.
The canonical notation of the answer will take the form 83 006 = 2 · 7 3 · 11 2.
Answer: 83 006 = 2 7 7 7 11 11 = 2 7 3 11 2.
Example 4
Factor the number 897,924,289.
Solution
To find the first prime factor, search through the prime numbers, starting with 2. The end of the search occurs at the number 937. Then p 1 = 937, a 1 = a: p 1 = 897 924 289: 937 = 958 297 and 897 924 289 = 937 958 297.
The second step of the algorithm is to iterate over smaller prime numbers. That is, we start with the number 937. The number 967 can be considered prime because it is a prime divisor of the number a 1 = 958,297. From here we get that p 2 = 967, then a 2 = a 1: p 1 = 958 297: 967 = 991 and 897 924 289 = 937 967 991.
The third step says that 991 is a prime number, since it does not have a single prime factor that does not exceed 991. The approximate value of the radical expression is 991< 40 2 . Иначе запишем как 991 < 40 2 . This shows that p 3 = 991 and a 3 = a 2: p 3 = 991: 991 = 1. We find that the decomposition of the number 897 924 289 into prime factors is obtained as 897 924 289 = 937 967 991.
Answer: 897 924 289 = 937 967 991.
Using divisibility tests for prime factorization
To factor a number into prime factors, you need to follow an algorithm. When there are small numbers, it is permissible to use the multiplication table and divisibility signs. Let's look at this with examples.
Example 5
If it is necessary to factorize 10, then the table shows: 2 · 5 = 10. The resulting numbers 2 and 5 are prime numbers, so they are prime factors for the number 10.
Example 6
If it is necessary to decompose the number 48, then the table shows: 48 = 6 8. But 6 and 8 are not prime factors, since they can also be expanded as 6 = 2 3 and 8 = 2 4. Then the complete expansion from here is obtained as 48 = 6 8 = 2 3 2 4. The canonical notation will take the form 48 = 2 4 · 3.
Example 7
When decomposing the number 3400, you can use the signs of divisibility. In this case, the signs of divisibility by 10 and 100 are relevant. From here we get that 3,400 = 34 · 100, where 100 can be divided by 10, that is, written as 100 = 10 · 10, which means that 3,400 = 34 · 10 · 10. Based on the divisibility test, we find that 3 400 = 34 10 10 = 2 17 2 5 2 5. All factors are prime. The canonical expansion takes the form 3 400 = 2 3 5 2 17.
When we find prime factors, we need to use divisibility tests and multiplication tables. If you imagine the number 75 as a product of factors, then you need to take into account the rule of divisibility by 5. We get that 75 = 5 15, and 15 = 3 5. That is, the desired expansion is an example of the form of the product 75 = 5 · 3 · 5.
If you notice an error in the text, please highlight it and press Ctrl+Enter
What does factoring mean? How to do it? What can you learn from factoring a number into prime factors? The answers to these questions are illustrated with specific examples.
Definitions:
A number that has exactly two different divisors is called prime.
A number that has more than two divisors is called composite.
Expand natural number to factor means to represent it as a product of natural numbers.
To factor a natural number into prime factors means to represent it as a product of prime numbers.
Notes:
- In the decomposition of a prime number, one of the factors is equal to one, and the other is equal to the number itself.
- It makes no sense to talk about factoring unity.
- A composite number can be factored into factors, each of which is different from 1.
Let's factor the number 150. For example, 150 is 15 times 10. 15 is a composite number. It can be factored into prime factors of 5 and 3. 10 is a composite number. It can be factored into prime factors of 5 and 2. By writing their decompositions into prime factors instead of 15 and 10, we obtained the decomposition of the number 150. |
|
|
The number 150 can be factorized in another way. For example, 150 is the product of the numbers 5 and 30. 5 is a prime number. 30 is a composite number. It can be thought of as the product of 10 and 3. 10 is a composite number. It can be factored into prime factors of 5 and 2. We obtained the factorization of 150 into prime factors in a different way. |
Note that the first and second expansions are the same. They differ only in the order of the factors. It is customary to write factors in ascending order. |
|
Every composite number can be factorized into prime factors in a unique way, up to the order of the factors. |
When factoring large numbers into prime factors, use column notation:
![]() |
The smallest prime number that is divisible by 216 is 2. Divide 216 by 2. We get 108. |
The resulting number 108 is divided by 2. Let's do the division. The result is 54. |
|
According to the test of divisibility by 2, the number 54 is divisible by 2. After dividing, we get 27. |
|
The number 27 ends with the odd digit 7. It Not divisible by 2. The next prime number is 3. Divide 27 by 3. We get 9. Least prime The number that 9 is divisible by is 3. Three is itself a prime number; it is divisible by itself and one. Let's divide 3 by ourselves. In the end we got 1. |
|
- A number is divisible only by those prime numbers that are part of its decomposition.
- A number is divisible only into those composite numbers whose decomposition into prime factors is completely contained in it.
Let's look at examples:
4900 is divisible by the prime numbers 2, 5 and 7 (they are included in the expansion of the number 4900), but is not divisible by, for example, 13. |
|
11 550 75. This is so because the decomposition of the number 75 is completely contained in the decomposition of the number 11550. The result of division will be the product of factors 2, 7 and 11. 11550 is not divisible by 4 because there is an extra two in the expansion of four. |
Find the quotient of dividing the number a by the number b, if these numbers are decomposed into prime factors as follows: a=2∙2∙2∙3∙3∙3∙5∙5∙19; b=2∙2∙3∙3∙5∙19
The decomposition of the number b is completely contained in the decomposition of the number a. |
|
The result of dividing a by b is the product of the three numbers remaining in the expansion of a. So the answer is: 30. |
Bibliography
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.
- Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium. 2006.
- Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - M.: Education, 1989.
- Rurukin A.N., Tchaikovsky I.V. Assignments for the mathematics course, grades 5-6. - M.: ZSh MEPhI, 2011.
- Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for 6th grade students at the MEPhI correspondence school. - M.: ZSh MEPhI, 2011.
- Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for 5-6 grades of secondary school. - M.: Education, Mathematics Teacher Library, 1989.
- Internet portal Matematika-na.ru ().
- Internet portal Math-portal.ru ().
Homework
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012. No. 127, No. 129, No. 141.
- Other tasks: No. 133, No. 144.
Every composite number can be uniquely represented as a product of prime factors. For example,
48 = 2 2 2 2 3, 225 = 3 3 5 5, 1050 = 2 3 5 5 7.
For small numbers this decomposition is easy is done on the basisMultiplication tables. For large numbers, we recommend using the following method, which we will consider using a specific example. Let's factorize the number 1463 into prime factors. To do this, use the table of prime numbers:
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.
We sort through the numbers in this table and stop at the number that is a divisor of this number. In our example, this is 7. Divide 1463 by 7 and get 209. Now we repeat the process of searching through prime numbers for 209 and stop at the number 11, which is its divisor (see). Divide 209 by 11 and get 19, which, according to the same table, is a prime number. Thus, we have:
In this article you will find all the necessary information to answer the question, how to factor a number into prime factors. First, a general idea of the decomposition of a number into prime factors is given, and examples of decompositions are given. The following shows the canonical form of decomposing a number into prime factors. After this, an algorithm is given for decomposing arbitrary numbers into prime factors and examples of decomposing numbers using this algorithm are given. Alternative methods are also considered that allow you to quickly factor small integers into prime factors using divisibility tests and multiplication tables.
Page navigation.
What does it mean to factor a number into prime factors?
First, let's look at what prime factors are.
It is clear that since the word “factors” is present in this phrase, then there is a product of some numbers, and the qualifying word “simple” means that each factor is a prime number. For example, in a product of the form 2·7·7·23 there are four prime factors: 2, 7, 7 and 23.
What does it mean to factor a number into prime factors?
This means that this number must be represented as a product of prime factors, and the value of this product must be equal to the original number. As an example, consider the product of three prime numbers 2, 3 and 5, it is equal to 30, thus the decomposition of the number 30 into prime factors is 2·3·5. Usually the decomposition of a number into prime factors is written as an equality; in our example it will be like this: 30=2·3·5. We emphasize separately that prime factors in the expansion can be repeated. This is clearly illustrated by the following example: 144=2·2·2·2·3·3. But a representation of the form 45=3·15 is not a decomposition into prime factors, since the number 15 is a composite number.
The following question arises: “What numbers can be decomposed into prime factors?”
In search of an answer to it, we present the following reasoning. Prime numbers, by definition, are among those greater than one. Considering this fact and , it can be argued that the product of several prime factors is a positive integer greater than one. Therefore, factorization into prime factors occurs only for positive integers that are greater than 1.
But can all integers greater than one be factored into prime factors?
It is clear that it is not possible to factor simple integers into prime factors. This is because prime numbers have only two positive factors - one and itself, so they cannot be represented as the product of two or more prime numbers. If the integer z could be represented as the product of prime numbers a and b, then the concept of divisibility would allow us to conclude that z is divisible by both a and b, which is impossible due to the simplicity of the number z. However, they believe that any prime number is itself a decomposition.
What about composite numbers? Are composite numbers decomposed into prime factors, and are all composite numbers subject to such decomposition? The fundamental theorem of arithmetic gives an affirmative answer to a number of these questions. The basic theorem of arithmetic states that any integer a that is greater than 1 can be decomposed into the product of prime factors p 1, p 2, ..., p n, and the decomposition has the form a = p 1 · p 2 ·… · p n, and this the expansion is unique, if you do not take into account the order of the factors
Canonical factorization of a number into prime factors
In the expansion of a number, prime factors can be repeated. Repeating prime factors can be written more compactly using . Let in the decomposition of a number the prime factor p 1 occur s 1 times, the prime factor p 2 – s 2 times, and so on, p n – s n times. Then the prime factorization of the number a can be written as a=p 1 s 1 ·p 2 s 2 ·…·p n s n. This form of recording is the so-called canonical factorization of a number into prime factors.
Let us give an example of the canonical decomposition of a number into prime factors. Let us know the decomposition 609 840=2 2 2 2 3 3 5 7 11 11, its canonical notation has the form 609 840=2 4 3 2 5 7 11 2.
The canonical factorization of a number into prime factors allows you to find all the divisors of the number and the number of divisors of the number.
Algorithm for factoring a number into prime factors
To successfully cope with the task of decomposing a number into prime factors, you need to have a very good knowledge of the information in the article prime and composite numbers.
The essence of the process of decomposing a positive integer number a that exceeds one is clear from the proof of the fundamental theorem of arithmetic. The point is to sequentially find the smallest prime divisors p 1, p 2, ..., p n of the numbers a, a 1, a 2, ..., a n-1, which allows us to obtain a series of equalities a=p 1 ·a 1, where a 1 = a:p 1 , a=p 1 ·a 1 =p 1 ·p 2 ·a 2 , where a 2 =a 1:p 2 , …, a=p 1 ·p 2 ·…·p n ·a n , where a n =a n-1:p n . When it turns out a n =1, then the equality a=p 1 ·p 2 ·…·p n will give us the desired decomposition of the number a into prime factors. It should also be noted here that p 1 ≤p 2 ≤p 3 ≤…≤p n.
It remains to figure out how to find the smallest prime factors at each step, and we will have an algorithm for decomposing a number into prime factors. A table of prime numbers will help us find prime factors. Let us show how to use it to obtain the smallest prime divisor of the number z.
We sequentially take prime numbers from the table of prime numbers (2, 3, 5, 7, 11, and so on) and divide the given number z by them. The first prime number by which z is evenly divided will be its smallest prime divisor. If the number z is prime, then its smallest prime divisor will be the number z itself. It should be recalled here that if z is not a prime number, then its smallest prime divisor does not exceed the number , where is from z. Thus, if among the prime numbers not exceeding , there was not a single divisor of the number z, then we can conclude that z is a prime number (more about this is written in the theory section under the heading This number is prime or composite).
As an example, we will show how to find the smallest prime divisor of the number 87. Let's take the number 2. Divide 87 by 2, we get 87:2=43 (remaining 1) (if necessary, see article). That is, when dividing 87 by 2, the remainder is 1, so 2 is not a divisor of the number 87. We take the next prime number from the prime numbers table, this is the number 3. Divide 87 by 3, we get 87:3=29. Thus, 87 is divisible by 3, therefore, the number 3 is the smallest prime divisor of the number 87.
Note that in the general case, to factor a number a into prime factors, we need a table of prime numbers up to a number not less than . We will have to refer to this table at every step, so we need to have it at hand. For example, to factorize the number 95 into prime factors, we will only need a table of prime numbers up to 10 (since 10 is greater than ). And to decompose the number 846,653, you will already need a table of prime numbers up to 1,000 (since 1,000 is greater than ).
We now have enough information to write down algorithm for factoring a number into prime factors. The algorithm for decomposing the number a is as follows:
- Sequentially sorting through the numbers from the table of prime numbers, we find the smallest prime divisor p 1 of the number a, after which we calculate a 1 =a:p 1. If a 1 =1, then the number a is prime, and it itself is its decomposition into prime factors. If a 1 is not equal to 1, then we have a=p 1 ·a 1 and move on to the next step.
- We find the smallest prime divisor p 2 of the number a 1 , to do this we sequentially sort through the numbers from the table of prime numbers, starting with p 1 , and then calculate a 2 =a 1:p 2 . If a 2 =1, then the required decomposition of the number a into prime factors has the form a=p 1 ·p 2. If a 2 is not equal to 1, then we have a=p 1 ·p 2 ·a 2 and move on to the next step.
- Going through the numbers from the table of prime numbers, starting with p 2, we find the smallest prime divisor p 3 of the number a 2, after which we calculate a 3 =a 2:p 3. If a 3 =1, then the required decomposition of the number a into prime factors has the form a=p 1 ·p 2 ·p 3. If a 3 is not equal to 1, then we have a=p 1 ·p 2 ·p 3 ·a 3 and move on to the next step.
- We find the smallest prime divisor p n of the number a n-1 by sorting through the prime numbers, starting with p n-1, as well as a n =a n-1:p n, and a n is equal to 1. This step is the last step of the algorithm; here we obtain the required decomposition of the number a into prime factors: a=p 1 ·p 2 ·…·p n.
For clarity, all the results obtained at each step of the algorithm for decomposing a number into prime factors are presented in the form of the following table, in which the numbers a, a 1, a 2, ..., a n are written sequentially in a column to the left of the vertical line, and to the right of the line - the corresponding smallest prime divisors p 1, p 2, ..., p n.
All that remains is to consider a few examples of the application of the resulting algorithm for decomposing numbers into prime factors.
Examples of prime factorization
Now we will look in detail examples of factoring numbers into prime factors. When decomposing, we will use the algorithm from the previous paragraph. Let's start with simple cases, and gradually complicate them in order to encounter all the possible nuances that arise when decomposing numbers into prime factors.
Example.
Factor the number 78 into its prime factors.
Solution.
We begin the search for the first smallest prime divisor p 1 of the number a=78. To do this, we begin to sequentially sort through prime numbers from the table of prime numbers. We take the number 2 and divide 78 by it, we get 78:2=39. The number 78 is divided by 2 without a remainder, so p 1 =2 is the first found prime divisor of the number 78. In this case, a 1 =a:p 1 =78:2=39. So we come to the equality a=p 1 ·a 1 having the form 78=2·39. Obviously, a 1 =39 is different from 1, so we move on to the second step of the algorithm.
Now we are looking for the smallest prime divisor p 2 of the number a 1 =39. We begin enumerating numbers from the table of prime numbers, starting with p 1 =2. Divide 39 by 2, we get 39:2=19 (remaining 1). Since 39 is not evenly divisible by 2, then 2 is not its divisor. Then we take the next number from the table of prime numbers (number 3) and divide 39 by it, we get 39:3=13. Therefore, p 2 =3 is the smallest prime divisor of the number 39, while a 2 =a 1:p 2 =39:3=13. We have the equality a=p 1 ·p 2 ·a 2 in the form 78=2·3·13. Since a 2 =13 is different from 1, we move on to the next step of the algorithm.
Here we need to find the smallest prime divisor of the number a 2 =13. In search of the smallest prime divisor p 3 of the number 13, we will sequentially sort through the numbers from the table of prime numbers, starting with p 2 =3. The number 13 is not divisible by 3, since 13:3=4 (rest. 1), also 13 is not divisible by 5, 7 and 11, since 13:5=2 (rest. 3), 13:7=1 (rest. 6) and 13:11=1 (rest. 2). The next prime number is 13, and 13 is divisible by it without a remainder, therefore, the smallest prime divisor p 3 of 13 is the number 13 itself, and a 3 =a 2:p 3 =13:13=1. Since a 3 =1, this step of the algorithm is the last, and the required decomposition of the number 78 into prime factors has the form 78=2·3·13 (a=p 1 ·p 2 ·p 3 ).
Answer:
78=2·3·13.
Example.
Express the number 83,006 as a product of prime factors.
Solution.
At the first step of the algorithm for decomposing a number into prime factors, we find p 1 =2 and a 1 =a:p 1 =83,006:2=41,503, from which 83,006=2·41,503.
At the second step, we find out that 2, 3 and 5 are not prime divisors of the number a 1 =41,503, but the number 7 is, since 41,503:7=5,929. We have p 2 =7, a 2 =a 1:p 2 =41,503:7=5,929. Thus, 83,006=2 7 5 929.
The smallest prime divisor of the number a 2 =5 929 is the number 7, since 5 929:7 = 847. Thus, p 3 =7, a 3 =a 2:p 3 =5 929:7 = 847, from which 83 006 = 2·7·7·847.
Next we find that the smallest prime divisor p 4 of the number a 3 =847 is equal to 7. Then a 4 =a 3:p 4 =847:7=121, so 83 006=2·7·7·7·121.
Now we find the smallest prime divisor of the number a 4 =121, it is the number p 5 =11 (since 121 is divisible by 11 and not divisible by 7). Then a 5 =a 4:p 5 =121:11=11, and 83 006=2·7·7·7·11·11.
Finally, the smallest prime divisor of the number a 5 =11 is the number p 6 =11. Then a 6 =a 5:p 6 =11:11=1. Since a 6 =1, this step of the algorithm for decomposing a number into prime factors is the last, and the desired decomposition has the form 83 006 = 2·7·7·7·11·11.
The result obtained can be written as the canonical decomposition of the number into prime factors 83 006 = 2·7 3 ·11 2.
Answer:
83 006=2 7 7 7 11 11=2 7 3 11 2 991 is a prime number. Indeed, it does not have a single prime divisor not exceeding ( can be roughly estimated as , since it is obvious that 991<40 2
), то есть, наименьшим делителем числа 991
является оно само. Тогда p 3 =991
и a 3 =a 2:p 3 =991:991=1
. Следовательно, искомое разложение числа 897 924 289
на простые множители имеет вид 897 924 289=937·967·991
.
Answer:
897 924 289=937·967·991.
Using divisibility tests for prime factorization
In simple cases, you can decompose a number into prime factors without using the decomposition algorithm from the first paragraph of this article. If the numbers are not large, then to decompose them into prime factors it is often enough to know the signs of divisibility. Let's give examples for clarification.
For example, we need to factor the number 10 into prime factors. From the multiplication table we know that 2·5=10, and the numbers 2 and 5 are obviously prime, so the prime factorization of the number 10 looks like 10=2·5.
Another example. Using the multiplication table, we will factor the number 48 into prime factors. We know that six is eight - forty-eight, that is, 48 = 6·8. However, neither 6 nor 8 are prime numbers. But we know that twice three is six, and twice four is eight, that is, 6=2·3 and 8=2·4. Then 48=6·8=2·3·2·4. It remains to remember that two times two is four, then we get the desired decomposition into prime factors 48 = 2·3·2·2·2. Let's write this expansion in canonical form: 48=2 4 ·3.
But when factoring the number 3,400 into prime factors, you can use the divisibility criteria. The signs of divisibility by 10, 100 allow us to state that 3400 is divisible by 100, with 3400=34·100, and 100 is divisible by 10, with 100=10·10, therefore, 3400=34·10·10. And based on the test of divisibility by 2, we can say that each of the factors 34, 10 and 10 is divisible by 2, we get 3 400=34 10 10=2 17 2 5 2 5. All factors in the resulting expansion are simple, so this expansion is the desired one. All that remains is to rearrange the factors so that they go in ascending order: 3 400 = 2·2·2·5·5·17. Let us also write down the canonical decomposition of this number into prime factors: 3 400 = 2 3 ·5 2 ·17.
When decomposing a given number into prime factors, you can use in turn both the signs of divisibility and the multiplication table. Let's imagine the number 75 as a product of prime factors. The test of divisibility by 5 allows us to state that 75 is divisible by 5, and we obtain that 75 = 5·15. And from the multiplication table we know that 15=3·5, therefore, 75=5·3·5. This is the required decomposition of the number 75 into prime factors.
Bibliography.
- Vilenkin N.Ya. and others. Mathematics. 6th grade: textbook for general education institutions.
- Vinogradov I.M. Fundamentals of number theory.
- Mikhelovich Sh.H. Number theory.
- Kulikov L.Ya. and others. Collection of problems in algebra and number theory: Textbook for students of physics and mathematics. specialties of pedagogical institutes.
Can be represented as a product of prime numbers.
Example. Let's represent the numbers 4, 6 and 8 as a product of prime factors:
The right-hand sides of the resulting equalities are called prime factorization.
This is a representation of a composite number as a product of prime factors.
Factor a composite number into prime factors- means to represent this number as a product of prime factors.
Prime factors in the expansion of a number can be repeated. Repeating prime factors can be written more compactly - in the form of a power.
Example.
24 = 2 2 2 3 = 2 3 3
Note. Prime factors are usually written in ascending order.
How to factor a number into prime factors
The sequence of actions when factoring a number into prime factors:
- We check the table of prime numbers to see if the given number is prime.
- If not, then we sequentially select the smallest prime number from the table of prime numbers by which this number is divisible without a remainder, and perform the division.
- We check using the table of prime numbers to see if the resulting quotient is a prime number.
- If not, then we sequentially select the smallest prime number from the table of prime numbers, by which the resulting quotient is divisible by a whole, and perform the division.
- We repeat points 3 and 4 until the quotient turns out to be one.
Example. Factor the number 102 into its prime factors.
Solution:
We begin the search for the smallest prime divisor of the number 102. To do this, we sequentially select the smallest prime number from the table of prime numbers, by which 102 will be divided without a remainder. We take the number 2 and try to divide 102 by it, we get:
The number 102 is divided by 2 without a remainder, so 2 is the first prime factor found. Since the dividend is equal to the divisor multiplied by the quotient, we can write:
Let's move on to the next step. We check using the table of prime numbers to see if the resulting quotient is a prime number. The number 51 is composite. Starting with the number 2, we select the smallest prime divisor of the number 51 from the table of prime numbers. The number 51 is not divisible by 2. We move on to the next number from the table of prime numbers (the number 3) and try to divide 51 by it, we get:
The number 51 is divided by 3, so 3 is the second prime factor found. Now we can represent the number 51 as a product. This process can be written like this:
102 = 2 51 = 2 3 17
We check using the table of prime numbers to see if the resulting quotient is a prime number. The number 17 is simple. This means that the smallest prime number that is divisible by 17 will be this number itself:
Since we got a unit in the quotient, the decomposition is complete. Thus, the decomposition of the number 102 into prime factors has the form:
102 = 2 3 17
Answer: 102 = 2 3 17.
In arithmetic, there is another form of notation that facilitates the process of decomposing composite numbers. It consists in recording the entire decomposition process in a column (in two columns separated by a vertical line). To the left of the vertical line, from top to bottom, write down sequentially: the given composite number, then the resulting quotients, and to the right of the line - the corresponding smallest prime factors.
Example. Factor the number 120 into prime factors.
Solution:
We write the number 120 and draw a vertical line to the right of it:
To the right of the line we write the smallest prime divisor of the number 120:
We perform the division and write the resulting quotient (60) under this number:
We select the smallest prime divisor for 60, write it to the right of the vertical line under the previous divisor and perform the division. We continue the process until we get a unit in the quotient:
In the quotient we have a unit, which means the decomposition is complete. After decomposing into a column, the factors should be written down in a line:
120 = 2 3 3 5.
Answer: 120 = 2 3 3 5.
A composite number can be factorized into its prime factors in a unique way.
This means that if, for example, the number 20 is decomposed into two twos and one five, then it will always decompose this way, regardless of whether we start the decomposition with small factors or with large ones. It is customary to start expansion with small factors, i.e., with twos, threes, etc.
New on the site | | | contact@site |
2018 − 2020 | website |