How to factor a three-digit number. Factorization - online calculator

(except 0 and 1) have at least two divisors: 1 and itself. Numbers that have no other divisors are called simple numbers. Numbers that have other divisors are called composite(or complex) numbers. There are an infinite number of prime numbers. The following are prime numbers 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 basic arithmetic operations, a binary mathematical operation in which one argument is added as many times as the other. In arithmetic, multiplication is a short form of adding a specified number of identical terms.

For example, the notation 5*3 means “add three fives,” that is, 5+5+5. The result of multiplication is called work, and the numbers to be multiplied are multipliers or factors. The first factor is sometimes called " multiplicand».

Every composite number can be factorized into prime factors. With any method, the same expansion is obtained, if you do not take into account the order in which the factors are written.

Factoring a number (Factorization).

Factorization (factorization)- enumeration of divisors - an algorithm for factorization or testing the primality of a number by completely enumerating all possible potential divisors.

Those., in simple language, factorization is the name given to the process of factoring numbers, expressed in scientific language.

The sequence of actions when factoring into prime factors:

1. Check whether the proposed number is prime.

2. If not, then, guided by the signs of division, we select a divisor from prime numbers, starting with the smallest (2, 3, 5 ...).

3. We repeat this action until the quotient is prime number.

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.

During decomposition large numbers For 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.
• The number is divisible only by those composite numbers, the decomposition of which into prime factors is completely contained in it.

Let's look at examples:

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

Bibliography

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.
2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium. 2006.
3. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - M.: Education, 1989.
4. Rurukin A.N., Tchaikovsky I.V. Assignments for the mathematics course for grades 5-6. - M.: ZSh MEPhI, 2011.
5. 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.
6. Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for grades 5-6 high school. - M.: Education, Mathematics Teacher Library, 1989.

1. Internet portal Matematika-na.ru ().
2. Internet portal Math-portal.ru ().

Homework

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012. No. 127, No. 129, No. 141.
2. Other tasks: No. 133, No. 144.

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?

It means that given 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 clearly illustrates next 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. Taking into account this fact and , it can be argued that the product of several prime factors is an integer positive number, exceeding 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 explained by the fact that prime numbers have only two positive divisors - one and itself, so they cannot be represented as a 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 general case To factor the 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 3,400 is divisible by 100, with 3,400=34·100, and 100 is divisible by 10, with 100=10·10, therefore, 3,400=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.

Any composite number can be factorized into prime factors. There can be several methods of decomposition. Either method produces the same result.

How to factor a number into prime factors in the most convenient way? Let's look at how best to do this using specific examples.

Examples. 1) Factor the number 1400 into prime factors.

1400 is divisible by 2. 2 is a prime number; there is no need to factor it. We get 700. Divide it by 2. We get 350. We also divide 350 by 2. The resulting number 175 can be divided by 5. The result is 35 - divide by 5 again. Total - 7. It can only be divided by 7. We get 1, division over.

The same number can be factorized differently:

It is convenient to divide 1400 by 10. 10 is not a prime number, so it needs to be factored into prime factors: 10=2∙5. The result is 140. We divide it again by 10=2∙5. We get 14. If 14 is divided by 14, then it should also be decomposed into a product of prime factors: 14=2∙7.

Thus, we again came to the same decomposition as in the first case, but faster.

Conclusion: when decomposing a number, it is not necessary to divide it only into prime factors. We divide by what is more convenient, for example, by 10. You just need to remember to decompose the compound divisors into simple factors.

2) Factor the number 1620 into prime factors.

The most convenient way to divide the number 1620 is by 10. Since 10 is not a prime number, we represent it as a product of prime factors: 10=2∙5. We got 162. It is convenient to divide it by 2. The result is 81. The number 81 can be divided by 3, but by 9 it is more convenient. Since 9 is not a prime number, we expand it as 9=3∙3. We get 9. We also divide it by 9 and expand it into the product of prime factors.

Factoring a large number is not an easy task. Most people have trouble figuring out four or five digit numbers. To make the process easier, write the number above the two columns.

• Let's factorize the number 6552.