Identical factors of decomposition of numbers. What are prime numbers? What does it mean to factor a number into prime factors?
Factoring a number into prime factors- This is a common problem that you need to be able to solve. Prime factorization may be needed when finding GCD (Greatest Common Factor) and LCM (Least Common Multiple), and when testing whether numbers are coprime.
All numbers can be divided into two main types:
- Prime number is a number that is divisible only by itself and 1.
- Composite number is a number that has divisors other than itself and 1.
To check whether a number is prime or composite, you can use a special table of prime numbers.
Prime numbers table
For ease of calculation, everything prime numbers were collected in a table. Below is a table of prime numbers from 1 to 1000.
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 | 211 | 223 |
227 | 229 | 233 | 239 | 241 | 251 | 257 | 263 | 269 | 271 | 277 | 281 |
283 | 293 | 307 | 311 | 313 | 317 | 331 | 337 | 347 | 349 | 353 | 359 |
367 | 373 | 379 | 383 | 389 | 397 | 401 | 409 | 419 | 421 | 431 | 433 |
439 | 443 | 449 | 457 | 461 | 463 | 467 | 479 | 487 | 491 | 499 | 503 |
509 | 521 | 523 | 541 | 547 | 557 | 563 | 569 | 571 | 577 | 587 | 593 |
599 | 601 | 607 | 613 | 617 | 619 | 631 | 641 | 643 | 647 | 653 | 659 |
661 | 673 | 677 | 683 | 691 | 701 | 709 | 719 | 727 | 733 | 739 | 743 |
751 | 757 | 761 | 769 | 773 | 787 | 797 | 809 | 811 | 821 | 823 | 827 |
829 | 839 | 853 | 857 | 859 | 863 | 877 | 881 | 883 | 887 | 907 | 911 |
919 | 929 | 937 | 941 | 947 | 953 | 967 | 971 | 977 | 983 | 991 | 997 |
Prime factorization
To factor a number into prime factors, you can use a table of prime numbers and signs of divisibility of numbers. Until the number becomes equal to 1, you need to select a prime number by which the current one is divided and perform the division. If it was not possible to find a single factor that is not equal to 1 and the number itself, then the number is prime. Let's look at how this is done with an example.
Factor the number 63140 into prime factors.
In order not to lose the factors, we will write them in a column, as shown in the picture. This solution is quite compact and convenient. Let's take a closer look at it.
Any composite number can be represented as a product of its prime divisors:
28 = 2 2 7
The right-hand sides of the resulting equalities are called prime factorization numbers 15 and 28.
To factor a given composite number into prime factors means to represent this number as a product of its prime factors.
The decomposition of a given number into prime factors is carried out as follows:
- First you need to select the smallest prime number from the table of prime numbers that divides the given composite number without a remainder, and perform the division.
- Next, you need to again select the smallest prime number by which the already obtained quotient will be divided without a remainder.
- The second action is repeated until one is obtained in the quotient.
As an example, let's factorize the number 940 into prime factors. Find the smallest prime number that divides 940. This number is 2:
Now we select the smallest prime number that is divisible by 470. This number is again 2:
The smallest prime number that is divisible by 235 is 5:
The number 47 is prime, which means that the smallest prime number that can be divided by 47 is the number itself:
Thus, we get the number 940, factored into prime factors:
940 = 2 470 = 2 2 235 = 2 2 5 47
If the decomposition of a number into prime factors resulted in several identical factors, then for brevity, they can be written in the form of a power:
940 = 2 2 5 47
It is most convenient to write decomposition into prime factors as follows: first we write down this composite number and draw a vertical line to the right of it:
To the right of the line we write the smallest prime divisor by which the given composite number is divided:
We perform the division and write the resulting quotient under the dividend:
We act with the quotient in the same way as with the given composite number, i.e., we select the smallest prime number by which it is divisible without a remainder and perform the division. And we repeat this until we get a unit in the quotient:
Please note that sometimes it can be quite difficult to factor a number into prime factors, since during the factorization we may encounter a large number that is difficult to immediately determine whether it is prime or composite. And if it is composite, then it is not always easy to find its smallest prime divisor.
Let’s try, for example, to factorize the number 5106 into prime factors:
Having reached the quotient 851, it is difficult to immediately determine its smallest divisor. We turn to the table of prime numbers. If there is a number in it that puts us in difficulty, then it is divisible only by itself and one. The number 851 is not in the table of prime numbers, which means it is composite. All that remains is to divide it by sequential search into prime numbers: 3, 7, 11, 13, ..., and so on until we find a suitable prime divisor. By brute force we find that 851 is divisible by the number 23.
Lesson in 6th grade on the topic
"Prime factorization"
Lesson objectives:
Educational:
Develop an understanding of the decomposition of numbers into prime factors, the ability to practically use the corresponding algorithm.
To develop skills in using divisibility signs when decomposing numbers into prime factors.
Educational:
Develop computational skills, the ability to generalize, analyze, identify patterns, and compare.
Educational:
To cultivate attention, a culture of mathematical thinking, and a serious attitude towards educational work.
Lesson content:
1. Oral counting.
2. Repetition of the material covered.
3. Explanation of new material.
4. Fixing the material.
5. Reflection.
6. Summing up the lesson.
During the classes
Motivation (self-determination) for educational activities.
Hello guys. The topic of our lesson is “Factoring numbers into prime factors.” You are already partially familiar with it. And in order to better set the goal of the lesson, we will work a little orally.
Follow the steps (orally) .
Calculate:
1. 15 x(325 -325) + 236x1 – 30:1 206
2. 207 – (0 x4376 -0:585) + 315: 315 208
3. (60 – 0:60) + (150:1 -48x0) 210
4. (707:707 +211x1):1 -0:123 212
Repetition of learned material
Continue the resulting row for 3 numbers
(206; 208;210; 212;214;216;218)
Choose divisible numbers from them
to: 2 (206; 208;210; 212;214;216;218)
by 3: (210;216)
at 9: (216)
at 5: (210)
by 4: (208; 212; 216)
Formulate the signs of divisibility
Questions: 1. What numbers are called prime?
2. What numbers are called composite?
3. What kind of number is 1?
4. Name all the prime numbers in the first two tens.
5. How many prime numbers are there in total?
6.Is the number 32 prime?
7.Is the number 73 prime?
Explanation of new material.
Let's solve a very interesting problem.
Once upon a time there were troubles and a grandmother. They had chicken Ryaba. The hen lays every seventh egg is golden, and every third is silver. Could this be possible?
(Answer: no, because 21 eggs can be gold or silver) Why?
What should we learn in class today? (Decompose any numbers into prime factors)
Why do you think we need this? (to solve more complex examples and also reduce fractions)
Today the topic of our lesson will help us better understand and solve such problems.
Solve the problem: You need to select a rectangular plot of land with an area of 18 square meters. m., What could be the dimensions of this area if they must be expressed in natural numbers?
Solution: 1. 18=1 x 18 = 2 x3 x3
2. 18= 2 x 9 = 2x3x3
3. 18=3 x 6 = 3 x2x 3
Work in pairs.
What have we done? (Presented as a product or factored). Is it possible to continue the decomposition? But as? What did you get?
Question: What can be said about these multipliers?
All factors are prime numbers.
Open the textbook What should I do? Who can explain to me how this is done? (Discussion in pairs)
Using the analyzed example, we will decompose the number 84 into prime factors (decomposition algorithm):
84 2 756 2 - the teacher shows on the board.
42 2 378 2
21 3 189 3 84 = 2x2∙3∙7 = 2 2 ∙3∙7
7 7 63 3
1 21 3 756= 2x2x3x3x3x3
Factor 756 into its prime factors. Compare with my solution. What did you notice?
On page 194, find the answer to the following question?
Any number can be expanded into a product of prime factors
the only way.
Reinforcing the material learned .
1. Factor the numbers into prime factors: 20; 188; 254.
we'll check Slide 12
20 2 188 2 254 2
10 2 94 2 127 127
5 5 47 47 1 1
1 1 1
№ 1. 20 = 2 2 ∙5; 188 = 2²∙47; 254 = 2∙127.
Everyone is offered cards. Students decide and check with the original, which is on the teacher’s desk. If done correctly, give yourself a plus sign in the summary table. (Solve by 3)
Card No. 2. Factor the numbers into prime factors: 30; 136; 438.
Card number 3. Factor the numbers into prime factors: 40; 125; 326.
Card No. 4. Factor the numbers into prime factors: 50; 78; 285.
Card No. 5. Factor the numbers into prime factors: 60; 654; 99.
Card number 6. Factor the numbers into prime factors: 70; 65; 136.
After the work is completed we will check.
№ 2. 30 = 2∙3∙5; 136 = 2 3 ∙17; 438 =2∙3∙73.
№3. 40 = 2 3 ∙5; 125 = 5 3 ; 326 = 2 ∙163
№4. 50 = 2∙5²; 78 = 2∙3∙13; 285 = 3∙5∙9.
№ 5. 60 = 2²∙3∙5; 654 = 2∙3∙109; 99 = 3²∙11
№ 6. 70 = 2∙5∙7; 65 = 5∙13; 136 = 2 3 ∙17.
Bottom line.
What does it mean to factor a number into prime factors?
(Expand natural number by prime factors - this means representing a number as a product of prime numbers.)
2) Is there a unique decomposition of a natural number into prime factors?
(No matter how we decompose a natural number into prime factors, we obtain its only decomposition; the order of the factors is not taken into account.)
Homework.
factor any 4 numbers into prime factors.
(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 main arithmetic operations, a binary mathematical operation in which one argument is added as many times as the other. In arithmetic, multiplication is understood as a short notation for 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.
Every natural number, except one, has two or more divisors. For example, the number 7 is divisible without a remainder only by 1 and 7, that is, it has two divisors. And the number 8 has divisors 1, 2, 4, 8, that is, as many as 4 divisors at once.
What is the difference between prime and composite numbers?
Numbers that have more than two divisors are called composite numbers. Numbers that have only two divisors: one and the number itself are called prime numbers.
The number 1 has only one division, namely the number itself. One is neither a prime nor a composite number.
- For example, the number 7 is prime and the number 8 is composite.
First 10 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. The number 2 is the only even prime number, all other prime numbers are odd.
The number 78 is composite, since in addition to 1 and itself, it is also divisible by 2. When divided by 2, we get 39. That is, 78 = 2*39. In such cases, they say that the number was factored into factors of 2 and 39.
Any composite number can be decomposed into two factors, each of which is greater than 1. This trick will not work with a prime number. So it goes.
Factoring a number into prime factors
As noted above, any composite number can be factorized into two factors. Let's take, for example, the number 210. This number can be decomposed into two factors 21 and 10. But the numbers 21 and 10 are also composite, let's decompose them into two factors. We get 10 = 2*5, 21=3*7. And as a result, the number 210 was decomposed into 4 factors: 2,3,5,7. These numbers are already prime and cannot be expanded. That is, we factored the number 210 into prime factors.
When factoring composite numbers into prime factors, they are usually written in ascending order.
It should be remembered that any composite number can be decomposed into prime factors and in a unique way, up to permutation.
- Usually, when decomposing a number into prime factors, divisibility criteria are used.
Let's factor the number 378 into prime factors
We will write down the numbers, separating them with a vertical line. The number 378 is divisible by 2, since it ends in 8. When divided, we get the number 189. The sum of the digits of the number 189 is divisible by 3, which means the number 189 itself is divisible by 3. The result is 63.
The number 63 is also divisible by 3, according to divisibility. We get 21, the number 21 can again be divided by 3, we get 7. Seven is divided only by itself, we get one. This completes the division. To the right after the line are the prime factors into which the number 378 is decomposed.
378|2
189|3
63|3
21|3
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