Expansion into powers of prime numbers. Factor. Factoring a number Factorization. Factoring a number into prime factors

Let's factor the number 120 into prime factors

120 = 2 ∙ 2 ∙ 2 ∙ 3 ∙ 5

Solution
Let's expand the number 120

120: 2 = 60
60: 2 = 30 - divisible by the prime number 2
30: 2 = 15 - divisible by the prime number 2
15: 3 = 5
We complete the division since 5 is a prime number

Answer: 120 = 2 ∙ 2 ∙ 2 ∙ 3 ​​∙ 5

Let's factor the number 246 into prime factors

246 = 2 ∙ 3 ∙ 41

Solution
Let's break down the number 246 into prime factors and highlight them in green. We begin to select a divisor from prime numbers, starting with the smallest prime number 2, until the quotient turns out to be a prime number

246: 2 = 123 - divisible by the prime number 2
123: 3 = 41 - divisible by the prime number 3.
We complete the division since 41 is a prime number

Answer: 246 = 2 ∙ 3 ​​∙ 41

Let's factor the number 1463 into prime factors

1463 = 7 ∙ 11 ∙ 19

Solution
Let's expand the number 1463 into prime factors and highlight them in green. We begin to select a divisor from prime numbers, starting with the smallest prime number 2, until the quotient turns out to be a prime number

1463: 7 = 209 - divisible by the prime number 7
209: 11 = 19
We complete the division since 19 is a prime number

Answer: 1463 = 7 ∙ 11 ∙ 19

Let's factor the number 1268 into prime factors

1268 = 2 ∙ 2 ∙ 317

Solution
Let's expand the number 1268 into prime factors and highlight them in green. We begin to select a divisor from prime numbers, starting with the smallest prime number 2, until the quotient turns out to be a prime number

1268: 2 = 634 - divisible by the prime number 2
634: 2 = 317 - divisible by the prime number 2.
We complete the division since 317 is a prime number

Answer: 1268 = 2 ∙ 2 ∙ 317

Let's factor the number 442464 into prime factors

442464

Solution
Let's expand the number 442464 into prime factors and highlight them in green. We begin to select a divisor from prime numbers, starting with the smallest prime number 2, until the quotient turns out to be a prime number

442464: 2 = 221232 - divisible by the prime number 2
221232: 2 = 110616 - divisible by the prime number 2
110616: 2 = 55308 - divisible by the prime number 2
55308: 2 = 27654 - divisible by the prime number 2
27654: 2 = 13827 - divisible by the prime number 2
13827: 3 = 4609 - divisible by the prime number 3
4609: 11 = 419 - divisible by the prime number 11.
We complete the division since 419 is a prime number

Answer: 442464 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 ∙ 11 ∙ 419

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 of 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 into prime factors 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. Note 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 a 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 also 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.

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(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.

That is, in simple terms, factorization is the name of 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 turns out to be a prime number.

What does factoring mean? How to do this? 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.

To factor a natural number 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.

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 within 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

References

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 5-6 grades of secondary 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.