Does the number 24 have a greatest multiple? Natural numbers (N). Prime and composite numbers. Divisor, multiple. Greatest common divisor, least common multiple. Least common multiple
The set of natural numbers = (1, 2, 3...). That is, the set of natural numbers is the set of all positive integers. The operations of addition, multiplication, subtraction and division are defined on natural numbers. The result of adding, multiplying and subtracting two natural numbers is a whole number. The result of dividing two natural numbers can be either an integer or a fraction.
For example: 20: 4 = 5 – the result of division is an integer.
20: 3 = 6 2/3 – the result of division is a fraction.
A natural number n is said to be divisible by a natural number m if the result of the division is an integer. In this case, the number m is called a divisor of the number n, and the number n is called a multiple of the number m.
In the first example, the number 20 is divisible by 4, 4 is a divisor of 20, and 20 is a multiple of 4.
In the second example, the number 20 is not divisible by the number 3; accordingly, there can be no question of divisors and multiples.
A number n is called prime if it has no divisors other than itself and one. Examples of prime numbers: 2, 7, 11, 97, etc.
A number n is called composite if it has divisors other than itself and one.
Any natural number can be decomposed into a product of primes, and this decomposition is unique, up to the order of the factors. For example: 36=2 2 3 3 = 2 3 2 3 = 3 2 3 2 – all these expansions differ only in the order of the factors.
The greatest common divisor of two numbers m and n is the largest natural number that is a divisor of both m and n. For example, the numbers 34 and 85 have a greatest common factor of 17.
The least common multiple of two numbers m and n is the smallest natural number that is a multiple of both m and n. For example, the numbers 15 and 4 have a least common multiple of 60.
A natural number, divisible by two prime numbers, is also divisible by their product. For example, if a number is divisible by 2 and 3, then it is divisible by 6 = 2 3, if by 11 and 7, then by 77.
Example: the number 6930 is divisible by 11 - 6930: 11 = 630, and is divisible by 7 - 6930: 7 = 990. We can safely say that this number is also divisible by 77. Let’s check: 6930: 77 = 90.
Algorithm for decomposing the number n into prime factors:
1. Find the smallest prime divisor of the number n (other than 1) - a1.
2. Divide the number n by a1, denoting the quotient as n1.
3. n=a1 n1.
4. We perform the same operation with n1 until we get a prime number.
Example: Factor the number 17,136 into prime factors
1. The smallest prime divisor other than 1, here 2.
2. 17 136: 2 = 8 568;
3. 17 136 = 8 568 2.
4. The smallest prime divisor of 8568 is 2.
5. 8 568: 2 = 4284;
6. 17 136 = 4284 2 2.
7. The smallest prime divisor of 4284 is 2.
8. 4284: 2 = 2142;
9. 17 136 = 2142 2 2 2.
10. The smallest prime divisor of 2142 is 2.
11. 2142: 2 = 1071;
12. 17 136 = 1071 2 2 2 2.
13. The smallest prime divisor of 1071 is 3.
14. 1071: 3 = 357;
15. 17 136 = 357 3 2 2 2 2.
16. The smallest prime divisor of 357 is 3.
17. 357: 3 = 119;
18. 17 136 = 119 3 3 2 2 2 2.
19. The smallest prime divisor of 119 is 7.
20. 119: 7 = 17;
21. 17 is a prime number, which means 17 136 = 17 7 3 3 2 2 2 2.
We have obtained the decomposition of the number 17,136 into prime factors.
Common multiples of natural numbersaAndbis a number that is a multiple of each of these numbers.
Smallest number of all common multiples A And b called least common multiple of these numbers.
Least common multiple of numbers A And b Let us agree to denote K( A, b).
For example, the two numbers 12 and 18 are common multiples of: 36, 72, 108, 144, 180, etc. The number 36 is the least common multiple of the numbers 12 and 18. You can write: K(12, 18) = 36.
For the least common multiple the following statements are true:
1. Least common multiple of numbers A And b
2. Least common multiple of numbers A And b no less than the larger of these numbers, i.e. If a >b, then K( A, b) ≥ A.
3. Any common multiple of numbers A And b divided by their least common multiple.
Greatest common divisor
The common divisor of natural numbers a andbis a number that is a divisor of each of these numbers.
The largest number of all common divisors of numbers A And b is called the greatest common divisor of these numbers.
Greatest common divisor of numbers A And b Let us agree to denote D( A, b).
For example, for the numbers 12 and 18, the common divisors are the numbers: 1, 2, 3, 6. The number 6 is 12 and 18. You can write: D(12, 18) = 6.
The number 1 is the common divisor of any two natural numbers a And b. If these numbers have no other common divisors, then D( A, b) = 1, and the numbers A And b are called mutually prime.
For example, the numbers 14 and 15 are relatively prime, since D(14, 15) = 1.
For the greatest common divisor the following statements are true:
1. Greatest common divisor of numbers a And b always exists and is unique.
2. Greatest common divisor of numbers A And b does not exceed the smaller of the given numbers, i.e. If a< b, That D(a, b) ≤ a.
3. Greatest common divisor of numbers a And b is divisible by any common divisor of these numbers.
Greatest common multiple of numbers A And b and their greatest common divisor are interrelated: the product of the least common multiple and the greatest common divisor of numbers A And b equal to the product of these numbers, i.e. K( a, b)·D( a, b) = a· b.
The following corollaries follow from this statement:
a) The least common multiple of two mutually prime numbers is equal to the product of these numbers, i.e. D( a, b) = 1 => K( a, b) = a· b;
For example, to find the least common multiple of the numbers 14 and 15, it is enough to multiply them, since D(14, 15) = 1.
b) A divided by the product of coprime numbers m And n, it is necessary and sufficient that it is divisible by m, and on n.
This statement is a sign of divisibility by numbers that can be represented as the product of two relatively prime numbers.
c) The quotients obtained by dividing two given numbers by their greatest common divisor are relatively prime numbers.
This property can be used when checking the correctness of the found greatest common divisor of given numbers. For example, let's check whether the number 12 is the greatest common divisor of the numbers 24 and 36. To do this, according to the last statement, we divide 24 and 36 by 12. We get the numbers 2 and 3, respectively, which are coprime. Therefore, D(24, 36)=12.
Problem 32. Formulate and prove the test for divisibility by 6.
Solution x divisible by 6, it is necessary and sufficient that it be divisible by 2 and 3.
Let the number x is divisible by 6. Then from the fact that x 6 and 62, it follows that x 2. And from the fact that x 6 and 63, it follows that x 3. We proved that in order for a number to be divisible by 6, it must be divisible by 2 and 3.
Let us show the sufficiency of this condition. Because x 2 and x 3, then x- common multiple of numbers 2 and 3. Any common multiple of numbers is divided by their least multiple, which means x K(2;3).
Since D(2, 3)=1, then K(2, 3)=2·3=6. Hence, x 6.
Problem 33. Formulate to 12, 15 and 60.
Solution. In order for a natural number x divisible by 12, it is necessary and sufficient that it be divisible by 3 and 4.
In order for a natural number x divisible by 15, it is necessary and sufficient that it be divisible by 3 and 5.
In order for a natural number x divisible by 60, it is necessary and sufficient that it be divisible by 4, 3 and 5.
Problem 34. Find numbers a And b, if K( a, b)=75, a· b=375.
Solution. Using the formula K( a,b)·D( a,b)=a· b, find the greatest common divisor of the required numbers A And b:
D( a, b) === 5.
Then the required numbers can be represented in the form A= 5R, b= 5q, Where p And q p and 5 q into equality a b= 275. Let's get 5 p·5 q=375 or p· q=15. We solve the resulting equation with two variables by selection: we find pairs of relatively prime numbers whose product is equal to 15. There are two such pairs: (3, 5) and (1, 15). Therefore, the required numbers A And b are: 15 and 25 or 5 and 75.
Problem 35. Find numbers A And b, if it is known that D( a, b) = 7 and a· b= 1470.
Solution. Since D( a, b) = 7, then the required numbers can be represented in the form A= 7R, b= 7q, Where p And q are mutually prime numbers. Let's substitute expressions 5 R and 5 q into equality a b = 1470. Then 7 p·7 q= 1470 or p· q= 30. We solve the resulting equation with two variables by selection: we find pairs of relatively prime numbers whose product is equal to 30. There are four such pairs: (1, 30), (2, 15), (3, 10), (5, 6). Therefore, the required numbers A And b are: 7 and 210, 14 and 105, 21 and 70, 35 and 42.
Problem 36. Find numbers A And b, if it is known that D( a, b) = 3 and A:b= 17:14.
Solution. Because a:b= 17:14, then A= 17R And b= 14p, Where R- greatest common divisor of numbers A And b. Hence, A= 17·3 = 51, b= 14·3 = 42.
Problem 37. Find numbers A And b, if it is known that K( a, b) = 180, a:b= 4:5.
Solution. Because a: b=4:5 then A=4R And b=5R, Where R- greatest common divisor of numbers a And b. Then R·180=4 R·5 R. Where R=9. Hence, a= 36 and b=45.
Problem 38. Find numbers A And b, if it is known that D( a,b)=5, K( a,b)=105.
Solution. Since D( a, b) K( a, b) = a· b, That a· b= 5 105 = 525. In addition, the required numbers can be represented in the form A= 5R And b= 5q, Where p And q are mutually prime numbers. Let's substitute expressions 5 R and 5 q into equality A· b= 525. Then 5 p·5 q=525 or p· q=21. We find pairs of relatively prime numbers whose product is equal to 21. There are two such pairs: (1, 21) and (3, 7). Therefore, the required numbers A And b are: 5 and 105, 15 and 35.
Problem 39. Prove that the number n(2n+ 1)(7n+ 1) is divisible by 6 for any natural n.
Solution. The number 6 is composite; it can be represented as the product of two relatively prime numbers: 6 = 2·3. If we prove that a given number is divisible by 2 and 3, then based on the test of divisibility by a composite number we can conclude that it is divisible by 6.
To prove that the number n(2n+ 1)(7n+ 1) is divisible by 2, we need to consider two possibilities:
1) n is divisible by 2, i.e. n= 2k. Then the product n(2n+ 1)(7n+ 1) will look like: 2 k(4k+ 1)(14k+ 1). This product is divisible by 2, because the first factor is divisible by 2;
2) n is not divisible by 2, i.e. n= 2k+ 1. Then the product n(2n+ 1 )(7n+ 1) will look like: (2 k+ 1)(4k+ 3)(14k+ 8). This product is divisible by 2, because the last factor is divisible by 2.
To prove that the work n(2n+ 1)(7n+ 1) is divisible by 3, three possibilities need to be considered:
1) n is divisible by 3, i.e. n= 3k. Then the product n(2n+ 1)(7n+ 1) will look like: 3 k(6k+ 1)(21k+ 1). This product is divisible by 3, because the first factor is divisible by 3;
2) n When divided by 3, the remainder is 1, i.e. n= 3k+ 1. Then the product n(2n+ 1)(7n+ 1) will look like: (3 k+ 1)(6k+ 3)(21k+ 8). This product is divisible by 3, because the second factor is divisible by 3;
3) n when divided by 3, the remainder is 2, i.e. n= 3k+ 2. Then the product n(2n+ 1)(7n+ 1) will look like: (3 k+ 2)(6k+ 5)(21k+ 15). This product is divisible by 3, because the last factor is divisible by 3.
So, it has been proven that the product n(2n+ 1)(7n+ 1) is divisible by 2 and 3. This means it is divisible by 6.
Exercises for independent work
1. Given two numbers: 50 and 75. Write down the set:
a) divisors of the number 50; b) divisors of the number 75; c) common divisors of given numbers.
What is the greatest common divisor of 50 and 75?
2. Is the number 375 a common multiple of the numbers: a) 125 and 75; b) 85 and 15?
3. Find numbers A And b, if it is known that K( a, b) = 105, a· b= 525.
4. Find numbers A And b, if it is known that D( a, b) = 7, a· b= 294.
5. Find numbers A And b, if it is known that D( a, b) = 5, a:b= 13:8.
6. Find numbers A And b, if it is known that K( a, b) = 224, a:b= 7:8.
7. Find numbers a And b, if it is known that D( a, b) = 3, K( a; b) = 915.
8. Prove the test for divisibility by 15.
9. From the set of numbers 1032, 2964, 5604, 8910, 7008, write down those that are divisible by 12.
10. Formulate the criteria for divisibility by 18, 36, 45, 75.
Key words of the summary:Integers. Arithmetic operations on natural numbers. Divisibility of natural numbers. Prime and composite numbers. Factoring a natural number into prime factors. Divisibility signs by 2, 3, 5, 9, 4, 25, 10, 11. Greatest common divisor (GCD), as well as least common multiple (LCD). Division with remainder.
Integers- these are numbers that are used to count objects - 1, 2, 3, 4 , ... But the number 0 is not natural!
The set of natural numbers is denoted by N. Record "3 ∈ N" means that the number three belongs to the set of natural numbers, and the notation "0 ∉ N" means that the number zero does not belong to this set.
Decimal number system- positional radix number system 10 .
Arithmetic operations on natural numbers
For natural numbers the following actions are defined: addition, subtraction, multiplication, division, exponentiation, root extraction. The first four actions are arithmetic.
Let a, b and c be natural numbers, then
1. ADDITION. Term + Term = Sum
Properties of addition
1. Communicative a + b = b + a.
2. Conjunctive a + (b + c) = (a + b) + c.
3. a + 0= 0 + a = a.
2. SUBTRACT. Minuend - Subtrahend = Difference
Properties of Subtraction
1. Subtracting the sum from the number a - (b + c) = a - b - c.
2. Subtracting a number from the sum (a + b) - c = a + (b - c); (a + b) - c = (a - c) + b.
3. a - 0 = a.
4. a - a = 0.
3. MULTIPLICATION. Multiplier * Multiplier = Product
Properties of Multiplication
1. Communicative a*b = b*a.
2. Conjunctive a*(b*c) = (a*b)*c.
3. 1 * a = a * 1 = a.
4. 0 * a = a * 0 = 0.
5. Distributive (a + b) * c = ac + bc; (a - b) * c = ac - bc.
4. DIVISION. Dividend: Divisor = Quotient
Properties of division
1. a: 1 = a.
2. a: a = 1. You can't divide by zero!
3. 0: a= 0.
Procedure
1. First of all, the actions in parentheses.
2. Then multiplication, division.
3. And only at the end addition and subtraction.
Divisibility of natural numbers. Prime and composite numbers.
Divisor of a natural number A is the natural number to which A divided without remainder. Number 1 is a divisor of any natural number.
The natural number is called simple, if it only has two divisor: one and the number itself. For example, the numbers 2, 3, 11, 23 are prime numbers.
A number that has more than two divisors is called composite. For example, the numbers 4, 8, 15, 27 are composite numbers.
Divisibility test works several numbers: if at least one of the factors is divisible by a certain number, then the product is also divisible by this number. Work 24 15 77 divided by 12 , since the multiplier of this number 24 divided by 12 .
Divisibility test for a sum (difference) numbers: if each term is divisible by a certain number, then the entire sum is divided by this number. If a: b And c:b, That (a + c) : b. And if a: b, A c not divisible by b, That a+c not divisible by a number b.
If a: c And c:b, That a: b. Based on the fact that 72:24 and 24:12, we conclude that 72:12.
Representation of a number as a product of powers of prime numbers is called factoring a number into prime factors.
Fundamental Theorem of Arithmetic: any natural number (except 1 ) or is simple, or it can be factorized in only one way.
When decomposing a number into prime factors, the signs of divisibility are used and the “column” notation is used. In this case, the divisor is located to the right of the vertical line, and the quotient is written under the dividend.
For example, task: factor a number into prime factors 330 . Solution:
Signs of divisibility into 2, 5, 3, 9, 10, 4, 25 and 11.
There are signs of divisibility into 6, 15, 45 etc., that is, into numbers whose product can be factorized 2, 3, 5, 9 And 10 .
Greatest common divisor
The largest natural number by which each of two given natural numbers is divisible is called greatest common divisor these numbers ( GCD). For example, GCD (10; 25) = 5; and GCD (18; 24) = 6; GCD (7; 21) = 1.
If the greatest common divisor of two natural numbers is equal to 1 , then these numbers are called mutually prime.
Algorithm for finding the greatest common divisor(NOD)
GCD is often used in problems. For example, 155 notebooks and 62 pens were divided equally between students in one class. How many students are there in this class?
Solution: Finding the number of students in this class comes down to finding the greatest common divisor of the numbers 155 and 62, since the notebooks and pens were divided equally. 155 = 5 31; 62 = 2 31. GCD (155; 62) = 31.
Answer: 31 students in the class.
Least common multiple
Multiples of a natural number A is a natural number that is divisible by A without a trace. For example, number 8 has multiples: 8, 16, 24, 32 , ... Any natural number has infinitely many multiples.
Least common multiple(LCM) is the smallest natural number that is a multiple of these numbers.
Algorithm for finding the least common multiple ( NOC):
LCM is also often used in problems. For example, two cyclists simultaneously started along a cycle track in the same direction. One makes a circle in 1 minute, and the other in 45 seconds. In what minimum number of minutes after the start of the movement will they meet at the start?
Solution: The number of minutes after which they will meet again at the start must be divided by 1 min, as well as on 45 s. In 1 min = 60 s. That is, it is necessary to find the LCM (45; 60). 45 = 32 5; 60 = 22 3 5. LCM (45; 60) = 22 32 5 = 4 9 5 = 180. The result is that the cyclists will meet at the start in 180 s = 3 min.
Answer: 3 min.
Division with remainder
If a natural number A is not divisible by a natural number b, then you can do division with remainder. In this case, the resulting quotient is called incomplete. The equality is fair:
a = b n + r,
Where A- divisible, b- divider, n- incomplete quotient, r- remainder. For example, let the dividend be equal 243 , divider - 4 , Then 243: 4 = 60 (remainder 3). That is, a = 243, b = 4, n = 60, r = 3, then 243 = 60 4 + 3 .
Numbers that are divisible by 2 without remainder, are called even: a = 2n, n ∈ N.
The remaining numbers are called odd: b = 2n + 1, n ∈ N.
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