# How to factor a number

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Online calculator "Prime Factorization"allows you to decompose any composite number into prime factors. To do this, you need to enter a number in the field and click the" Calculate "button. The peculiarity of this calculator is that it will not only give an answer, but also provide a detailed solution. Using our calculator, you You can quickly get the result, and a detailed solution will help you figure out how the calculation was made.

All natural numbers can be divided into two groups of numbers: simple and compound.

Prime number Is a number that have only two divisors (unit and this number itself), i.e. is divided without remainder only into one and into itself. It is generally accepted that unit (1) is not a prime number. Example of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc. There are an infinite number of primes, the table below shows primes up to 1000.

Composite number Is the number that have more than two divisors. Any composite number can be represented as a product of primes, for example: 84 = 2 · 2 · 3 · 7.

## How to factor the number?

At school in mathematics, factorization is usually written in columns (in two columns). This is done like this: in the left column we write out the original number, then

• We take the smallest prime number - 2 and by the signs of divisibility or by regular division we check whether the original number is divisible by 2.
• If it is divisible, then we write 2. In the right column, then divide the original number by 2 and write the result in the left column under the original number.
• If it is not divisible, then we take the following prime number - 3.

We repeat these steps, while working with the last number in the left column and with the current prime number. The decomposition ends when the number 1 is written in the left column.

To better understand the algorithm, let’s take a look at a few examples.

Decision. Write the number 84 in the left column:

We take the first prime number - two and check if 84 is divisible by 2. Since 84 ends in 4 and 4 divides by 2, then 84 is divisible by 2 based on divisibility. We write 2 in the right column. 84: 2 = 42, the number 42 is written in the left column. We got this:

Now we are already working with the number 42. The number 42 is divided by 2, so we write 2 in the right column, 42: 2 = 21, the number 21 is written in the left column.

The number 21 is not divisible by 2, so we check its divisibility by the next prime number - 3. The number 21 is divisible by 3, 21: 3 = 7. We recorded 3 in the right column, 7 in the left. Got

The number 7 is a prime number, so in the right column we write 7, in the left we write 1. As a result, we got:

That's it, the number is laid out!

As a result, all the prime factors of 84 appeared in the right column. That is, 84 = 2 ∙ 2 ∙ 3 ​​∙ 7.

## Theory of factorization of numbers.

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What does it mean to factorize? How to do it? What can I find out by decomposing the number into prime factors? The answers to these questions are illustrated by specific examples.

If you have difficulty understanding the topic, we recommend that you look at the lesson “Number as an object of study (Number Theory)”

## What does it mean to decompose a number into prime factors?

A prime is a number that has exactly two different divisors.

A compound is a number that has more than two divisors.

Factoring a natural number means representing it as a product of natural numbers.

To decompose a natural number into prime factors means to represent it as a product of primes.

• In the expansion of a prime number, one of the factors is equal to one, and the other is equal to this very number.
• Talking about factoring a unit does not make sense.
• A composite number can be factorized, each of which is different from 1.

## Example. Factorization of 150

Factor 150. For example, 150 is 15 times 10.

15 is a composite number. It can be decomposed into prime factors 5 and 3.

10 is a composite number. It can be decomposed into prime factors 5 and 2.

Writing instead of 15 and 10 of their factorization, we get a factorization of 150.

The number 150 can be factorized differently. For example, 150 is the product of the numbers 5 and 30.

5 - a prime number.

30 is a composite number. It can be represented as a product of 10 and 3.

10 is a composite number. It can be decomposed into prime factors 5 and 2.

We got a prime factorization of 150 in another way.

Note that the first and second decompositions are the same. They differ only in the order of the factors.

It is customary to record factors in ascending order.

Any composite number can be decomposed into prime factors uniquely up to the order of factors.

## Example. Factor 216

When decomposing large numbers into prime factors, they use a column entry:

 The smallest prime number divided by 216 is 2.Divide 216 by 2. Get 108.The resulting number 108 is divided by 2.Perform the division. We get 54 as a result.According to the sign of divisibility by 2, the number 54 is divided by 2.Having completed the division, we get 27.The number 27 ends with an odd number 7. ItNot divisible by 2. The next prime number is 3.Divide 27 by 3. Get 9. Least SimpleThe number divided by 9 is 3. Three is itself a prime number, it is divided into itself and by one. Divide 3 into yourself. As a result, we got 1.

## Relationship between the divisibility of a compound number and its factorization

• A number is divided only by those prime numbers that are part of its decomposition.
• The number is divided only by those composite numbers, the decomposition of which into prime factors is completely contained in it.

4900 is divisible by primes 2, 5 and 7. (they are included in the decomposition of 4900), but it is not divisible, for example, by 13.

1.

2.

.

11 550 75. This is so because the decomposition of 75 is completely contained in the decomposition of 11550.

The result of the division will be the product of the factors 2, 7 and 11.

11550 is not divided by 4 because there is an extra deuce in the decomposition of four.

Find the quotient of dividing a by b if these numbers are factorized as follows a = 2 ∙ 2 ∙ 2 ∙ 3 ​​∙ 3 ∙ 3 ∙ 5 ∙ 5 ∙ 19, b = 2 ∙ 2 ∙ 3 ​​∙ 3 ∙ 5 ∙ 19

The decomposition of b is completely contained in the expansion of a.

The result of dividing a by b is the product of the three numbers remaining in the decomposition of a.

Bibliography

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Schwarzburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.
2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics grade 6. - Gymnasium. 2006.
3. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a math textbook. - M .: Education, 1989.
4. Rurukin A.N., Tchaikovsky I.V. Tasks on the course of mathematics 5-6 grade. - M.: ZH MEPhI, 2011.
5. Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5–6. A manual for students of 6 classes of the correspondence school of MEPhI. - M.: ZH MEPhI, 2011.
6. Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Interlocutor textbook for grades 5–6 of high school. - M .: Education, Library of the teacher of mathematics, 1989.

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

Homework

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

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