Synopsis keywords:Integers. Arithmetic operations on natural numbers. Divisibility of natural numbers. Prime and composite numbers. Decomposition of a natural number into prime factors. Signs of divisibility by 2, 3, 5, 9, 4, 25, 10, 11. The greatest common divisor (GCD), as well as the least common multiple (LCM). Division with remainder.
Integers 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 N. Recording "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 number system based on 10 .
Arithmetic operations on natural numbers
For natural numbers, the following actions are defined: addition, subtraction, multiplication, division, exponentiation, root extraction. The first four steps are arithmetic.
Let a, b and c be natural numbers, then
1. ADDITION. Term + Term = Sum
Addition properties
1. Commutative a + b = b + a.
2. Combinative a + (b + c) \u003d (a + b) + c.
3. a + 0= 0 + a = a.
2. SUBTRACT. Reduced - Subtracted = Difference
subtraction properties
1. Subtraction of the sum from the number a - (b + c) \u003d a - b - c.
2. Subtracting a number from the sum (a + b) - c \u003d a + (b - c); (a + b) - c \u003d (a - c) + b.
3. a - 0 = a.
4. a - a \u003d 0.
3. MULTIPLICATION. Multiplier * Multiplier = Product
Multiplication Properties
1. Commutative a * b \u003d b * a.
2. Combinative a * (b * c) \u003d (a * b) * c.
3. 1 * a = a * 1 = a.
4. 0 * a = a * 0 = 0.
5. Distribution (a + b) * c \u003d ac + bc; (a - b) * c \u003d ac - bc.
4. DIVISION. Dividend: Divisor = Quotient
division properties
1. a: 1 = a.
2. a: a = 1. You can't divide by zero!
3. 0: a=0.
Procedure
1. First of all, actions in brackets.
2. Then multiplication, division.
3. And only at the end of addition, subtraction.
Divisibility of natural numbers. Prime and composite numbers.
Divisor of a natural number a is called the natural number by 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 with more than two divisors is called composite. For example, the numbers 4, 8, 15, 27 are composite numbers.
divisibility sign works several numbers: if at least one of the factors is divisible by some number, then the product is also divisible by this number. Work 24 15 77 divided by 12 , since the factor of this number 24 divided by 12 .
Sign of divisibility of the sum (difference) numbers: if each term is divisible by some number, then the whole sum is divisible by this number. If a a:b and c:b, then (a + c) : b. And if a:b, a c not divisible by b, then a+c not divisible by number b.
If a a:c and c:b, then a:b. Based on the fact that 72:24 and 24:12, we conclude that 72:12.
The representation of a number as a product of powers of prime numbers is called decomposing a number into prime factors.
Fundamental theorem of arithmetic: any natural number (except 1 ) or is simple, or it can be decomposed into prime factors in only one way.
When decomposing a number into prime factors, divisibility signs are used and the “column” notation is used. In this case, the divisor is located to the right of the vertical bar, and the quotient is written under the dividend.
For example, the task: decompose a number into prime factors 330 . Decision:
Signs of divisibility by 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 factored 2, 3, 5, 9 and 10 .
Greatest Common Divisor
The largest natural number by which each of the 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 1 , then these numbers are called coprime.
Algorithm for Finding the Greatest Common Divisor(GCD)
GCD is often used in problems. For example, 155 notebooks and 62 pens were divided equally between students of the same class. How many students are in this class?
Decision: Finding the number of students in this class is reduced to finding the greatest common divisor of the numbers 155 and 62, since notebooks and pens were divided equally. 155 = 531; 62 = 231. GCD (155; 62) = 31.
Answer: 31 students in the class.
Least common multiple
Multiple 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.
The algorithm for finding the least common multiple ( NOC):
LCM is also often used in problems. For example, two cyclists started at the same time on the cycle track in the same direction. One makes a circle in 1 min, and the other in 45 s. In what least number of minutes after the start of the movement will they meet at the start?
Decision: The number of minutes after which they meet again at the start must be divisible 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 = 325; 60 = 22 3 5. NOC (45; 60) = 22 32 5 = 4 9 5 = 180. As a result, it turns out that the cyclists will meet at the start after 180 s = 3 min.
Answer: 3 min.
Division with remainder
If a natural number a not divisible by a natural number b, then you can do division with remainder. In this case, the resulting quotient is called incomplete. The right equality is:
a = b n + r,
where a- divisible b- divider, n- incomplete quotient, r- remainder. For example, let the dividend be 243 , divider - 4 , then 243: 4 = 60 (remainder 3). That is, a \u003d 243, b \u003d 4, n \u003d 60, r \u003d 3, then 243 = 60 4 + 3 .
Numbers that are divisible by 2 without a trace, are called even: a = 2n,n ∈ N.
The rest of the numbers are called odd: b = 2n + 1,n ∈ N.
This is a synopsis on the topic. "Integers. Signs of Divisibility». To continue, select next steps:
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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 addition, multiplication and subtraction of two natural numbers is an integer. And the result of dividing two natural numbers can be either an integer or a fractional number.
For example: 20: 4 = 5 - the result of division is an integer.
20: 3 \u003d 6 2/3 - the result of division is a fractional number.
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 20 is divisible by 4, 4 is a divisor of 20 , 20 is a multiple of 4.
In the second example, the number 20 is not divisible by the number 3, so 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 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 factors.
The greatest common divisor of two numbers m and n is the largest natural number that is a divisor of both m and a divisor of n. For example, for the numbers 34 and 85, the greatest common divisor is 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, for the numbers 15 and 4, the least common multiple would be 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 also divisible by 6 = 23, if by 11 and by 7, then by 77.
Example: the number 6930 is divisible by 11 - 6930: 11 \u003d 630, and is divisible by 7 - 6930: 7 \u003d 990. We can safely say that this number is also divisible by 77. Let's check: 6930: 77 \u003d 90.
Algorithm for decomposing the number n into prime factors:
1. Find the smallest prime divisor of n (other than 1) - a1.
2. Divide the number n by a1, denote the quotient by n1.
3. n=a1 n1.
4. We do the same operation with n1 until we get a prime number.
Example: Factoring the number 17,136 into prime factors
1. The smallest prime divisor other than 1 is 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, so 17 136 = 17 7 3 3 2 2 2 2.
We have obtained a decomposition of the number 17,136 into prime factors.
common multiple of natural numbersaandbis a number that is a multiple of each of the given numbers.
The smallest number of all common multiples a and b called the least common multiple of these numbers.
Least common multiple of numbers a and b let us denote K( a, b).
For example, two numbers 12 and 18 are common multiples: 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) \u003d 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 not less than the larger of the given numbers, i.e. if a >b, then K( a, b) ≥ a.
3. Any common multiple of numbers a and b is divisible by their least common multiple.
Greatest Common Divisor
Common divisor of natural numbers a andbis the number that is the divisor of each of the given numbers.
The largest number of all common divisors of numbers a and b is called the greatest common divisor of the given numbers.
Greatest Common Divisor of Numbers a and b let us 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 a 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 called coprime.
For example, the numbers 14 and 15 are coprime 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 smallest of the given numbers, i.e. if a< b, then 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 related: the product of the least common multiple and the greatest common divisor of numbers a and b is equal to the product of these numbers, i.e. K( a, b)D( a, b) = a· b.
The consequences follow from this statement:
a) The least common multiple of two relatively 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 divisible 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, which can be represented as a product of two coprime numbers.
c) The quotients obtained by dividing two given numbers by their greatest common divisor are coprime 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.
Task 32. Formulate and prove the test for divisibility by 6.
Decision x is 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 have 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. As x 2 and x 3, then x- the common multiple of the numbers 2 and 3. Any common multiple of the numbers is divisible by their smallest multiple, which means x K(2;3).
Since D(2, 3)=1, then K(2, 3)=2 3=6. Hence, x 6.
Task 33. Formulate at 12, 15 and 60.
Decision. In order for a natural number x is divisible by 12, it is necessary and sufficient that it be divisible by 3 and 4.
In order for a natural number x is divisible by 15, it is necessary and sufficient that it be divisible by 3 and 5.
In order for a natural number x is divisible by 60, it is necessary and sufficient that it be divisible by 4, 3, and 5.
Task 34. Find numbers a and b, if K( a, b)=75, a· b=375.
Decision. Using the formula K( a,b)D( a,b)=a· b, we find the greatest common divisor of the desired numbers a and b:
D( a, b) === 5.
Then the desired numbers can be represented as a= 5R, b= 5q, where p and q p and 5 q into equality a b= 275. Get 5 p·5 q=375 or p· q=15. We solve the resulting equation with two variables by selection: we find pairs of coprime numbers whose product is equal to 15. There are two such pairs: (3, 5) and (1, 15). Therefore, the desired numbers a and b these are: 15 and 25 or 5 and 75.
Task 35. Find numbers a and b, if it is known that D( a, b) = 7 and a· b= 1470.
Decision. Since D( a, b) = 7, then the desired numbers can be represented as a= 7R, b= 7q, where p and q are relatively prime numbers. 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 coprime numbers whose product is equal to 30. There are four such pairs: (1, 30), (2, 15), (3, 10), (5, 6). Therefore, the desired numbers a and b these are: 7 and 210, 14 and 105, 21 and 70, 35 and 42.
Task 36. Find numbers a and b, if it is known that D( a, b) = 3 and a:b= 17:14.
Decision. As 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.
Decision. As 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.
Decision. Since D( a, b) K( a, b) = a· b, then a· b= 5 105 = 525. In addition, the desired numbers can be represented as a= 5R and b= 5q, where p and q are relatively prime numbers. 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 coprime numbers whose product is equal to 21. There are two such pairs: (1, 21) and (3, 7). Therefore, the desired numbers a and b these are: 5 and 105, 15 and 35.
Task 39. Prove that the number n(2n+ 1)(7n+ 1) is divisible by 6 for any natural n.
Decision. The number 6 is composite, it can be represented as a product of two coprime numbers: 6 = 2 3. If we prove that a given number is divisible by 2 and 3, then, based on the test for 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, there are two possibilities to consider:
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 must 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, it gives a remainder of 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 is proved that the product n(2n+ 1)(7n+ 1) is divisible by 2 and 3. So it is divisible by 6.
Exercises for independent work
1. Two numbers are given: 50 and 75. Write down the set:
a) divisors of the number 50; b) divisors of the number 75; c) common divisors of these 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 out those that are divisible by 12.
10. Formulate signs of divisibility by 18, 36, 45, 75.
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