To simplify the division of natural numbers, rules for dividing into the numbers of the first ten and the numbers 11, 25 were derived, which are combined into the section signs of divisibility of natural numbers. Below are the rules by which the analysis of a number without dividing it by another natural number will answer the question, is a natural number a multiple of the numbers 2, 3, 4, 5, 6, 9, 10, 11, 25 and the digit unit?
Natural numbers that have digits (ending in) 2,4,6,8,0 in the first digit are called even.
Divisibility test for numbers by 2
All even natural numbers are divisible by 2, for example: 172, 94.67, 838, 1670.
Divisibility test for numbers by 3
All natural numbers whose sum of digits is divisible by 3 are divisible by 3. For example:
39 (3 + 9 = 12; 12: 3 = 4);
16 734 (1 + 6 + 7 + 3 + 4 = 21; 21:3 = 7).
Divisibility test for numbers by 4
All natural numbers are divisible by 4, the last two digits of which are zeros or a multiple of 4. For example:
124 (24: 4 = 6);
103 456 (56: 4 = 14).
Divisibility test for numbers by 5
Divisibility test for numbers by 6
Those natural numbers that are divisible by 2 and 3 at the same time are divisible by 6 (all even numbers that are divisible by 3). For example: 126 (b - even, 1 + 2 + 6 = 9, 9: 3 = 3).
Divisibility test for numbers by 9
Those natural numbers whose sum of digits is a multiple of 9 are divisible by 9. For example:
1179 (1 + 1 + 7 + 9 = 18, 18: 9 = 2).
Divisibility test for numbers by 10
Divisibility test for numbers by 11
Only those natural numbers are divisible by 11 for which the sum of the digits occupying even places is equal to the sum of the digits occupying odd places, or the difference between the sum of the digits of odd places and the sum of the digits of even places is a multiple of 11. For example:
105787 (1 + 5 + 8 = 14 and 0 + 7 + 7 = 14);
9,163,627 (9 + 6 + b + 7 = 28 and 1 + 3 + 2 = 6);
28 — 6 = 22; 22: 11 = 2).
Divisibility test for numbers by 25
Divide by 25 are those natural numbers whose last two digits are zeros or are a multiple of 25. For example:
2 300; 650 (50: 25 = 2);
1 475 (75: 25 = 3).
Sign of divisibility of numbers by digit unit
Those natural numbers whose number of zeros is greater than or equal to the number of zeros of the digit unit are divided into a digit unit. For example: 12,000 is divisible by 10, 100 and 1000.
The series of articles on divisibility criteria continues test of divisibility by 3. This article first gives a formulation of the test for divisibility by 3, and gives examples of using this test to find out which of the given integers are divisible by 3 and which are not. Below is a proof of the test for divisibility by 3. Also considered are approaches to establishing divisibility by 3 of numbers given as the value of some expression.
Page navigation.
Test for divisibility by 3, examples
Let's start with formulations of the test of divisibility by 3: an integer is divisible by 3 if the sum of its digits is divisible by 3, but if the sum of the digits of a given number is not divisible by 3, then the number itself is not divisible by 3.
From the above formulation it is clear that the test of divisibility by 3 cannot be used without the ability to perform the addition of natural numbers. Also, to successfully apply the test of divisibility by 3, you need to know that of all single-digit natural numbers, the numbers 3, 6 and 9 are divisible by 3, but the numbers 1, 2, 4, 5, 7 and 8 are not divisible by 3.
Now we can consider the simplest examples of using the test of divisibility by 3. Let's find out whether the number is divisible by 3? 42. To do this, we calculate the sum of the digits of the number?42, it is equal to 4+2=6. Since 6 is divisible by 3, then, due to the divisibility test by 3, we can say that the number ?42 is also divisible by 3. But the positive integer 71 is not divisible by 3, since the sum of its digits is 7+1=8, and 8 is not divisible by 3.
Is 0 divisible by 3? To answer this question, you won’t need the property of divisibility by 3; here you need to remember the corresponding property of divisibility, which states that zero is divisible by any integer. So 0 is divisible by 3.
In some cases, to show that a given number has or does not have the ability to be divisible by 3, the test of divisibility by 3 must be used several times in a row. Let's give an example.
Show that the number 907,444,812 is divisible by 3.
The sum of the digits of the number 907 444 812 is 9+0+7+4+4+4+8+1+2=39. To find out whether 39 is divisible by 3, let's calculate its sum of digits: 3+9=12. And to find out whether 12 is divisible by 3, we find the sum of the digits of the number 12, we have 1+2=3. Since we received the number 3, which is divisible by 3, then, by virtue of the divisibility test by 3, the number 12 is divisible by 3. Therefore, 39 is divisible by 3, since the sum of its digits is 12, and 12 is divisible by 3. Finally, 907,333,812 is divisible by 3, since the sum of its digits is 39, and 39 is divisible by 3.
To consolidate the material, we will analyze the solution to another example.
Is the number divisible by 3?543205?
Let's calculate the sum of the digits of this number: 5+4+3+2+0+5=19. In turn, the sum of the digits of the number 19 is 1+9=10, and the sum of the digits of the number 10 is 1+0=1. Since we received the number 1, which is not divisible by 3, from the test of divisibility by 3 it follows that 10 is not divisible by 3. Therefore, 19 is not divisible by 3, since the sum of its digits is 10, and 10 is not divisible by 3. Therefore, the original number?543205 is not divisible by 3, since the sum of its digits, equal to 19, is not divisible by 3.
It is worth noting that directly dividing a given number by 3 also allows us to conclude whether a given number is divisible by 3 or not. By this we want to say that we should not neglect division in favor of the criterion of divisibility by 3. In the last example, dividing 543,205 by 3 with a column, we would make sure that 543,205 is not evenly divisible by 3, from which we could say that ?543,205 is not divisible by 3.
Proof of the test of divisibility by 3
The following representation of the number a will help us prove the test of divisibility by 3. We can expand any natural number a into digits, after which the rule of multiplication by 10, 100, 1,000 and so on allows us to obtain a representation of the form a=a n ·10 n +a n?1 ·10 n?1 +…+a 2 ·10 2 +a 1 ·10+a 0, where a n, a n?1, ..., a 0 are the numbers from left to right in the notation of the number a. For clarity, we give an example of such a representation: 528=500+20+8=5·100+2·10+8.
Now let’s write down a number of fairly obvious equalities: 10=9+1=3·3+1, 100=99+1=33·3+1, 1 000=999+1=333·3+1 and so on.
Substituting into the equality a=a n ·10 n +a n?1 ·10 n?1 +…+a 2 ·10 2 +a 1 ·10+a 0 instead of 10, 100, 1 000 and so on the expressions 3·3+1 , 33·3+1 , 999+1=333·3+1 and so on, we get
.
The properties of addition of natural numbers and the properties of multiplication of natural numbers allow the resulting equality to be rewritten as follows:
Expression 
is the sum of the digits of the number a. For brevity and convenience, let us denote it by the letter A, that is, we accept . Then we obtain a representation of the number a of the form that we will use to prove the test for divisibility by 3.
Also, to prove the test for divisibility by 3, we need the following properties of divisibility:
- For an integer a to be divisible by an integer b it is necessary and sufficient that the modulus of the number a is divisible by the modulus of the number b;
- if in the equality a=s+t all terms except one are divisible by some integer b, then this one term is also divisible by b.
Now we are fully prepared and can carry out proof of divisibility by 3, for convenience, we formulate this criterion in the form of a necessary and sufficient condition for divisibility by 3.
For an integer a to be divisible by 3, it is necessary and sufficient that the sum of its digits is divisible by 3.
For a=0 the theorem is obvious.
If a is non-zero, then the modulus of the number a is a natural number, then the representation is possible, where is the sum of the digits of the number a.
Since the sum and product of integers is an integer, then it is an integer, then, by the definition of divisibility, the product is divisible by 3 for any a 0, a 1, ..., a n.
If the sum of the digits of a number a is divisible by 3, that is, A is divisible by 3, then, due to the divisibility property indicated before the theorem, it is divisible by 3, therefore, a is divisible by 3. So the sufficiency is proven.
If a is divisible by 3, then it is also divisible by 3, then, due to the same property of divisibility, the number A is divisible by 3, that is, the sum of the digits of the number a is divisible by 3. The necessity has been proven.
Other cases of divisibility by 3
Sometimes integers are not specified explicitly, but as the value of some expression with a variable given the value of the variable. For example, the value of an expression for some natural number n is a natural number. It is clear that when specifying numbers in this way, direct division by 3 will not help to establish their divisibility by 3, and the test of divisibility by 3 cannot always be applied. Now we will look at several approaches to solving such problems.
The essence of these approaches is to represent the original expression as a product of several factors, and if at least one of the factors is divisible by 3, then, due to the corresponding divisibility property, it will be possible to conclude that the entire product is divisible by 3.
Sometimes Newton's binomial allows one to implement this approach. Let's look at the example solution.
Is the value of the expression divisible by 3 for any natural number n?
Equality is obvious. Let's use Newton's binomial formula: 
In the last expression, we can take 3 out of brackets, and we will get. The resulting product is divided by 3, since it contains a factor of 3, and the value of the expression in brackets for natural n represents a natural number. Therefore, it is divisible by 3 for any natural number n.
In many cases, the method of mathematical induction allows one to prove divisibility by 3. Let's look at its application when solving an example.
Prove that for any natural number n, the value of the expression is divisible by 3.
To prove this, we will use the method of mathematical induction.
When n=1, the value of the expression is , and 6 is divided by 3.
Suppose that the value of the expression is divisible by 3 when n=k, that is, divisible by 3.
Considering that it is divisible by 3, we will show that the value of the expression for n=k+1 is divisible by 3, that is, we will show that 
divisible by 3.
Let's make some transformations: 
The expression is divisible by 3 and the expression 
is divisible by 3, so their sum is divisible by 3.
So, using the method of mathematical induction, divisibility by 3 for any natural number n was proven.
Let's show another approach to proving divisibility by 3. If we show that for n=3 m, n=3 m+1 and n=3 m+2, where m is an arbitrary integer, the value of some expression (with variable n) is divisible by 3, then this will prove Divisibility of an expression by 3 for any integer n. Let's consider this approach when solving the previous example.
Show what is divisible by 3 for any natural number n.
For n=3·m we have. The resulting product is divisible by 3, since it contains a factor of 3, divisible by 3.
The resulting product is also divisible by 3.
And this product is divisible by 3.
Therefore, it is divisible by 3 for any natural number n.
In conclusion, we present the solution to another example.
Is the value of the expression divisible by 3? 
for some natural number n.
For n=1 we have. The sum of the digits of the resulting number is 3, so the test of divisibility by 3 allows us to say that this number is divisible by 3.
For n=2 we have. The sum of the digits and this number is 3, so it is divisible by 3.
It is clear that for any other natural number n we will have numbers whose sum of digits is 3, therefore, these numbers are divisible by 3.
Thus, for any natural number n is divisible by 3.
www.cleverstudents.ru
Mathematics, 6th grade, textbook for students of general education organizations, Zubareva I.I., Mordkovich A.G., 2014
Mathematics, 6th grade, textbook for students of general education organizations, Zubareva I.I., Mordkovich A.G., 2014.
The theoretical material in the textbook is presented in such a way that the teacher can apply a problem-based approach to teaching. Using a notation system, exercises of four difficulty levels are distinguished. In each paragraph, control tasks are formulated based on what students should know and be able to do in order to achieve the level of the standard of mathematical education. At the end of the textbook there are home tests and answers. Color illustrations (drawings and diagrams) provide a high level of clarity of educational material.
Complies with the requirements of the Federal State Educational Standard LLC.
Tasks.
4. Draw a triangle ABC and mark point O outside it (as in Figure 11). Construct a figure symmetrical to triangle ABC relative to point O.
5. Draw a triangle KMN and construct a figure symmetrical to this triangle with respect to:
a) its vertices are points M;
b) point O - the middle of side MN.
6. Construct a symmetrical figure:
a) ray OM relative to point O; write down which point is symmetrical to point O;
b) ray OM relative to an arbitrary point A not belonging to this ray;
c) line AB relative to point O not belonging to this line;
d) line AB relative to point O belonging to this line; write down which point is symmetrical to point O.
In each case, characterize the relative position of the centrally symmetrical figures.
Table of contents
Chapter I. Positive and negative numbers. Coordinates
§ 1. Rotation and central symmetry
§ 2. Positive and negative numbers. Coordinate line
§ 3. Modulus of a number. Opposite numbers
§ 4. Comparison of numbers
§ 5. Parallelism of lines
§ 6. Numerical expressions containing the signs “+”, “-”
§ 7. Algebraic sum and its properties
§ 8. The rule for calculating the value of the algebraic sum of two numbers
§ 9. Distance between points of a coordinate line
§ 10. Axial symmetry
§ 11. Numerical intervals
§ 12. Multiplication and division of positive and negative numbers
§ 13. Coordinates
§ 14. Coordinate plane
§ 15. Multiplication and division of ordinary fractions
§ 16. Multiplication rule for combinatorial problems
Chapter II. Converting literal expressions
§ 17. Expanding parentheses
§ 18. Simplification of expressions
§ 19. Solution of equations
§ 20. Solving problems involving composing equations
§ 21. Two main problems on fractions
§ 22. Circle. Circumference
§ 23. Circle. Area of a circle
§ 24. Ball. Sphere
Chapter III. Divisibility of natural numbers
§ 25. Divisors and multiples
§ 26. Divisibility of a product
§ 27. Divisibility of the sum and difference of numbers
§ 28. Tests for divisibility by 2, 5, 10, 4 and 25
§ 29. Tests for divisibility by 3 and 9
§ 30. Prime numbers. Factoring a number into prime factors
§ 31. Greatest common divisor
§ 32. Coprime numbers. Divisibility test for a product. Least common multiple
Chapter IV. Mathematics around us
§ 33. The ratio of two numbers
§ 34. Diagrams
§ 35. Proportionality of quantities
§ 36. Solving problems using proportions
§ 37. Miscellaneous tasks
§ 38. First acquaintance with the concept of “probability”
§ 39. First acquaintance with calculating probability
Home tests
Topics for project activities
Answers
Download the e-book for free in a convenient format and read:
Mathematics
REFERENCE MATERIAL ON MATHEMATICS FOR GRADES 1-6.
Dear parents! If you are looking for a math tutor for your child, then this ad is for you. I offer Skype tutoring: preparation for the Unified State Exam, Unified State Exam, closing knowledge gaps. Your benefits are obvious:
1) Your child is at home, and you can be calm about him;
2) Classes are held at a time convenient for the child, and you can even attend these classes. I explain it simply and clearly on the usual school board.
3) You can think of other important advantages of Skype lessons yourself!
Write to me at: or immediately add me to Skype, and we will agree on everything. Prices are affordable.
P.S. Classes are possible in groups of 2-4 students.
Sincerely, Tatyana Yakovlevna Andryushchenko is the author of this site.
Dear friends!
I am pleased to invite you to download free math reference materials for 5th grade. Download here!
Dear friends!
It's no secret that some children have difficulty with multiplication and long division. Most often this is due to insufficient knowledge of the multiplication table. I suggest you learn your multiplication tables using lotto. See more details here. Download lotto here.
Dear friends! You will soon be faced (or have already faced) with the need to decide percent problems. Such problems begin to be solved in the 5th grade and are completed. but they don’t finish solving problems involving percentages! These tasks are found both in tests and in exams: both transfer ones and the Unified State Exam and Unified State Exam. What to do? We need to learn to solve such problems. My book “How to Solve Percentage Problems” will help you with this. Details here!
Adding numbers.
- a+b=c, where a and b are terms, c is the sum.
- To find the unknown term, you need to subtract the known term from the sum.
Subtracting numbers.
- a-b=c, where a is the minuend, b is the subtrahend, c is the difference.
- To find the unknown minuend, you need to add the subtrahend to the difference.
- To find the unknown subtrahend, you need to subtract the difference from the minuend.
Multiplying numbers.
- a·b=c, where a and b are factors, c is the product.
- To find an unknown factor, you need to divide the product by the known factor.
Dividing numbers.
- a:b=c, where a is the dividend, b is the divisor, c is the quotient.
- To find the unknown dividend, you need to multiply the divisor by the quotient.
- To find an unknown divisor, you need to divide the dividend by the quotient.
Laws of addition.
- a+b=b+a(commutative: rearranging the terms does not change the sum).
- (a+b)+c=a+(b+c)(combinative: in order to add a third number to the sum of two terms, you can add the sum of the second and third to the first number).
Addition table.
- 1+9=10; 2+8=10; 3+7=10; 4+6=10; 5+5=10; 6+4=10; 7+3=10; 8+2=10; 9+1=10.
- 1+19=20; 2+18=20; 3+17=20; 4+16=20; 5+15=20; 6+14=20; 7+13=20; 8+12=20; 9+11=20; 10+10=20; 11+9=20; 12+8=20; 13+7=20; 14+6=20; 15+5=20; 16+4=20; 17+3=20; 18+2=20; 19+1=20.
Laws of multiplication.
- a·b=b·a(commutative: rearranging the factors does not change the product).
- (a b) c=a (b c)(combinative: to multiply the product of two numbers by a third number, you can multiply the first number by the product of the second and third).
- (a+b)c=ac+bc(distributive law of multiplication relative to addition: in order to multiply the sum of two numbers by a third number, you can multiply each term by this number and add the resulting results).
- (a-b) c=a c-b c(distributive law of multiplication relative to subtraction: in order to multiply the difference of two numbers by a third number, you can multiply the minuend and subtract by this number separately and subtract the second from the first result).
Multiplication table.
2·1=2; 3·1=3; 4·1=4; 5·1=5; 6·1=6; 7·1=7; 8·1=8; 9·1=9.
2·2=4; 3·2=6; 4·2=8; 5·2=10; 6·2=12; 7·2=14; 8·2=16; 9·2=18.
2·3=6; 3·3=9; 4·3=12; 5·3=15; 6·3=18; 7·3=21; 8·3=24; 9·3=27.
2·4=8; 3·4=12; 4·4=16; 5·4=20; 6·4=24; 7·4=28; 8·4=32; 9·4=36.
2·5=10; 3·5=15; 4·5=20; 5·5=25; 6·5=30; 7·5=35; 8·5=40; 9·5=45.
2·6=12; 3·6=18; 4·6=24; 5·6=30; 6·6=36; 7·6=42; 8·6=48; 9·6=54.
2·7=14; 3·7=21; 4·7=28; 5·7=35; 6·7=42; 7·7=49; 8·7=56; 9·7=63.
2·8=16; 3·8=24; 4·8=32; 5·8=40; 6·8=48; 7·8=56; 8·8=64; 9·8=72.
2·9=18; 3·9=27; 4·9=36; 5·9=45; 6·9=54; 7·9=63; 8·9=72; 9·9=81.
2·10=20; 3·10=30; 4·10=40; 5·10=50; 6·10=60; 7·10=70; 8·10=80; 9·10=90.
Divisors and multiples.
- Divider natural number A name the natural number to which A divided without remainder. (The numbers 1, 2, 3, 4, 6, 8, 12, 24 are divisors of the number 24, since 24 is divisible by each of them without a remainder) 1 is the divisor of any natural number. The greatest divisor of any number is the number itself.
- Multiples natural number b is a natural number that is divisible by b. (The numbers 24, 48, 72,... are multiples of the number 24, since they are divisible by 24 without a remainder). The smallest multiple of any number is the number itself.
Divisibility criteria for natural numbers.
- The numbers used when counting objects (1, 2, 3, 4,...) are called natural numbers. The set of natural numbers is denoted by the letter N.
- Numbers 0, 2, 4, 6, 8 called even in numbers. Numbers that end in even digits are called even numbers.
- Numbers 1, 3, 5, 7, 9 called odd in numbers. Numbers that end in odd digits are called odd numbers.
- Test for divisibility by number 2. All natural numbers ending in an even digit are divisible by 2.
- Test for divisibility by number 5. All natural numbers ending in 0 or 5 are divisible by 5.
- Divisibility test for the number 10. All natural numbers ending in 0 are divisible by 10.
- Test for divisibility by number 3. If the sum of the digits of a number is divisible by 3, then the number itself is divisible by 3.
- Divisibility test for the number 9. If the sum of the digits of a number is divisible by 9, then the number itself is divisible by 9.
- Test for divisibility by number 4. If a number made up of the last two digits of a given number is divisible by 4, then the given number itself is divisible by 4.
- Divisibility test for the number 11. If the difference between the sum of the digits in odd places and the sum of the digits in even places is divisible by 11, then the number itself is divisible by 11.
- A prime number is a number that has only two divisors: one and the number itself.
- A number that has more than two divisors is called composite.
- The number 1 is neither a prime number nor a composite number.
- Writing a composite number as a product of only prime numbers is called factoring a composite number into prime factors. Any composite number can be uniquely represented as a product of prime factors.
- The greatest common divisor of given natural numbers is the largest natural number by which each of these numbers is divided.
- The greatest common divisor of these numbers is equal to the product of the common prime factors in the expansions of these numbers. Example. GCD(24, 42)=2·3=6, since 24=2·2·2·3, 42=2·3·7, their common prime factors are 2 and 3.
- If natural numbers have only one common divisor - one, then these numbers are called relatively prime.
- The least common multiple of given natural numbers is the smallest natural number that is a multiple of each of the given numbers. Example. LCM(24, 42)=168. This is the smallest number that is divisible by both 24 and 42.
- To find the LCM of several given natural numbers, you need to: 1) decompose each of the given numbers into prime factors; 2) write out the decomposition of the larger number and multiply it by the missing factors from the decomposition of other numbers.
- The least multiple of two relatively prime numbers is equal to the product of these numbers.
b- the denominator of the fraction shows how many equal parts it is divided into;
a-the numerator of the fraction shows how many such parts were taken. The fraction bar means the division sign.
Sometimes instead of a horizontal fractional line they put an oblique line, and an ordinary fraction is written like this: a/b.
- U proper fraction the numerator is less than the denominator.
- U improper fraction the numerator is greater than the denominator or equal to the denominator.
If the numerator and denominator of a fraction are multiplied or divided by the same natural number, you get an equal fraction.
Dividing both the numerator and denominator of a fraction by their common divisor other than one is called reducing the fraction.
- A number consisting of an integer part and a fractional part is called a mixed number.
- To represent an improper fraction as a mixed number, you need to divide the numerator of the fraction by the denominator, then the incomplete quotient will be the integer part of the mixed number, the remainder will be the numerator of the fractional part, and the denominator will remain the same.
- To represent a mixed number as an improper fraction, you need to multiply the integer part of the mixed number by the denominator, add the numerator of the fractional part to the resulting result and write it in the numerator of the improper fraction, leaving the denominator the same.
- Ray Oh with the starting point at the point ABOUT, on which are indicated single cut to and direction, called coordinate beam.
- The number corresponding to the point of the coordinate ray is called coordinate this point. For example , A(3). Read: point A with coordinate 3.
- Lowest common denominator ( NCD) of these irreducible fractions is the least common multiple ( NOC) denominators of these fractions.
- To reduce fractions to the least common denominator, you need to: 1) find the least common multiple of the denominators of the given fractions, it will be the least common denominator. 2) find an additional factor for each fraction by dividing the new denominator by the denominator of each fraction. 3) multiply the numerator and denominator of each fraction by its additional factor.
- Of two fractions with the same denominator, the one with the larger numerator is greater, and the one with the smaller numerator is smaller.
- Of two fractions with the same numerators, the one with the smaller denominator is greater, and the one with the larger denominator is smaller.
- To compare fractions with different numerators and different denominators, you must reduce the fractions to their lowest common denominator and then compare fractions with the same denominators.
Operations on ordinary fractions.
- To add fractions with the same denominators, you need to add their numerators and leave the denominator the same.
- If you need to add fractions with different denominators, first reduce the fractions to the lowest common denominator, and then add the fractions with the same denominators.
- To subtract fractions with like denominators, subtract the numerator of the second fraction from the numerator of the first fraction, leaving the denominator the same.
- If you need to subtract fractions with different denominators, then they are first brought to a common denominator, and then fractions with the same denominators are subtracted.
- When performing addition or subtraction operations on mixed numbers, these operations are performed separately for the whole parts and for the fractional parts, and then the result is written as a mixed number.
- The product of two ordinary fractions is equal to a fraction whose numerator is equal to the product of the numerators, and the denominator is equal to the product of the denominators of these fractions.
- To multiply a common fraction by a natural number, you need to multiply the numerator of the fraction by this number, but leave the denominator the same.
- Two numbers whose product is equal to one are called reciprocal numbers.
- When multiplying mixed numbers, they are first converted to improper fractions.
- To find a fraction of a number, you need to multiply the number by that fraction.
- To divide a common fraction by a common fraction, you need to multiply the dividend by the reciprocal of the divisor.
- When dividing mixed numbers, they are first converted into improper fractions.
- To divide a common fraction by a natural number, you need to multiply the denominator of the fraction by this natural number, and leave the numerator the same. ((2/7):5=2/(7·5)=2/35).
- To find a number by its fraction, you need to divide the number corresponding to it by this fraction.
- A decimal fraction is a number written in the decimal system and having digits less than one. (3.25; 0.1457, etc.)
- The places after the decimal point in a decimal fraction are called decimal places.
- The decimal will not change if you add or remove zeros at the end of the decimal.
To add decimal fractions, you need to: 1) equalize the number of decimal places in these fractions; 2) write them down one after the other so that the comma is written under the comma; 3) perform the addition, not paying attention to the comma, and put a comma in the sum under the commas in the added fractions.
To subtract decimal fractions, you need to: 1) equalize the number of decimal places in the minuend and the subtrahend; 2) sign the subtrahend under the minuend so that the comma is under the comma; 3) perform the subtraction, not paying attention to the comma, and in the resulting result place a comma under the commas of the minuend and the subtrahend.
- To multiply a decimal fraction by a natural number, you need to multiply it by this number, ignoring the comma, and in the resulting product, separate as many digits to the right with a comma as there were after the decimal point in this fraction.
- To multiply one decimal fraction by another, you need to perform the multiplication, not paying attention to the commas, and in the resulting result, separate as many digits on the right with a comma as there were after the decimal points in both factors together.
- To multiply a decimal fraction by 10, 100, 1000, etc., you need to move the decimal point to the right by 1, 2, 3, etc. digits.
- To multiply a decimal by 0.1; 0.01; 0.001, etc. you need to move the decimal point to the left by 1, 2, 3, etc. digits.
- To divide a decimal fraction by a natural number, you need to divide the fraction by this number, as natural numbers are divided, and put a comma in the quotient when the division of the whole part is completed.
- To divide a decimal fraction by 10, 100, 1000, etc., you need to move the decimal point to the left by 1, 2, 3, etc. digits.
- To divide a number by a decimal fraction, you need to move the commas in the dividend and divisor as many digits to the right as there are after the decimal point in the divisor, and then divide by the natural number.
- To divide a decimal by 0.1; 0.01; 0.001, etc., you need to move the decimal point to the right by 1, 2, 3, etc. digits. (Dividing a decimal by 0.1, 0.01, 0.001, etc. is the same as multiplying that decimal by 10, 100, 1000, etc.)
To round a number to any digit, we underline the digit of this digit, and then we replace all the digits after the underlined one with zeros, and if they are after the decimal point, we discard them. If the first digit replaced by a zero or discarded is 0, 1, 2, 3 or 4, then the underlined digit is left unchanged. If the first digit replaced by a zero or discarded is 5, 6, 7, 8 or 9, then the underlined digit is increased by 1.
The arithmetic mean of several numbers.
The arithmetic mean of several numbers is the quotient of dividing the sum of these numbers by the number of terms.
The range of a number of numbers.
The difference between the largest and smallest values of a data series is called the range of the series of numbers.
Mode of number series.
The number that occurs with the highest frequency among the given numbers in a series is called the mode of the number series.
- One hundredth part is called a percentage. Purchase a book that teaches “How to Solve Percentage Problems.”
- To express percentages as a fraction or a natural number, you need to divide the percentage by 100%. (4%=0.04; 32%=0.32).
- To express a number as a percentage, you need to multiply it by 100%. (0.65=0.65·100%=65%; 1.5=1.5·100%=150%).
- To find the percentage of a number, you need to express the percentage as a common or decimal fraction and multiply the resulting fraction by the given number.
- To find a number by its percentage, you need to express the percentage as an ordinary or decimal fraction and divide the given number by this fraction.
- To find what percentage the first number is from the second, you need to divide the first number by the second and multiply the result by 100%.
- The quotient of two numbers is called the ratio of these numbers. a:b or a/b– the ratio of numbers a and b, and a is the previous term, b is the next term.
- If the members of a given relation are rearranged, then the resulting relation is called the inverse of the given relation. The relationships b/a and a/b are mutually inverse.
- The ratio will not change if both terms of the ratio are multiplied or divided by the same number other than zero.
- The equality of two ratios is called proportion.
- a:b=c:d. This is a proportion. Read: A this applies to b, How c refers to d. The numbers a and d are called the extreme terms of the proportion, and the numbers b and c are called the middle terms of the proportion.
- The product of the extreme terms of a proportion is equal to the product of its middle terms. For proportion a:b=c:d or a/b=c/d the main property is written like this: a·d=b·c.
- To find the unknown extreme term of a proportion, you need to divide the product of the middle terms of the proportion by the known extreme term.
- To find the unknown middle term of a proportion, you need to divide the product of the extreme terms of the proportion by the known middle term. Proportion problems.
Let the value y depends on the size X. If when increasing X several times the size at increases by the same amount, then such values X And at are called directly proportional.
If two quantities are directly proportional, then the ratio of two arbitrarily taken values of the first quantity is equal to the ratio of two corresponding values of the second quantity.
The ratio of the length of a segment on a map to the length of the corresponding distance on the ground is called the map scale.
Let the value at depends on the size X. If when increasing X several times the size at decreases by the same amount, then such values X And at are called inversely proportional.
If two quantities are inversely proportional, then the ratio of two arbitrarily taken values of one quantity is equal to the inverse ratio of the corresponding values of the other quantity.
- A set is a collection of some objects or numbers, composed according to some general properties or laws (many letters on a page, many proper fractions with a denominator of 5, many stars in the sky, etc.).
- Sets consist of elements and can be finite or infinite. A set that does not contain a single element is called the empty set and is denoted by O.
- A bunch of IN called a subset of a set A, if all elements of the set IN are elements of the set A.
- Intersection of sets A And IN is a set whose elements belong to the set A and many IN.
- Union of sets A And IN is a set whose elements belong to at least one of these sets A And IN.
Lots of numbers.
- N– set of natural numbers: 1, 2, 3, 4,…
- Z– a set of integers: …, -4, -3, -2, -1, 0, 1, 2, 3, 4,…
- Q– a set of rational numbers representable as a fraction m/n, Where m– whole, n– natural (-2; 3/5; v9; v25, etc.)
- A coordinate line is a straight line on which a positive direction, a reference point (point O) and a unit segment are given.
- Each point on the coordinate line corresponds to a certain number, which is called the coordinate of this point. For example, A(5). They read: point A with coordinate five. AT 3). They read: point B with coordinate minus three.
- Modulus of the number a (write |a|) call the distance from the origin to the point corresponding to a given number A. The modulus of any number is non-negative. |3|=3; |-3|=3, because the distance from the origin to the number -3 and to the number 3 is equal to three unit segments. |0|=0 .
- By definition of the modulus of a number: |a|=a, If a?0 And |a|=-a, If a b.
- If, when comparing numbers a and b, the difference a-b is a negative number, then a , then they are called strict inequalities.
- If inequalities are written by signs? or?, then they are called non-strict inequalities.
Properties of numerical inequalities.
G) 
Inequality of the form x?a. Answer:
The use of divisibility skills simplifies calculations and proportionately increases the speed of their execution. Let us examine in detail the main characteristic features of divisibility.
The most straightforward test of divisibility for units: all numbers are divided by one. It’s just as elementary with the signs of divisibility by two, five, ten. You can divide by two an even number or one whose final digit is 0, by five - a number whose final digit is 5 or 0. Only those numbers with a final digit of 0 can be divided by ten. 100 - only those numbers whose two final digits are zeros, on 1000 - only those with three trailing zeros.
For example:
The number 79516 can be divided by 2, since it ends in 6—an even number; 9651 is not divisible by 2, since 1 is an odd number; 1790 is divisible by 2 since the final digit is zero. 3470 is divisible by 5 (the final digit is 0); 1054 is not divisible by 5 (the final digit is 4). 7800 is divisible by 10 and 100; 542000 is divisible by 10, 100, 1000.
Less widely known, but very convenient to use, are characteristic features of divisibility on 3 And 9 , 4 , 6 And 8, 25 . There are also characteristic features of divisibility into 7, 11, 13, 17, 19 and so on, but they are used much less frequently in practice.
A characteristic feature of division by 3 and 9.
On three and/or on nine Those numbers whose result of adding digits is a multiple of three and/or nine will be divided without a remainder.
For example:
The number 156321, the result of addition 1 + 5 + 6 + 3 + 2 + 1 = 18 is divisible by 3 and divisible by 9, respectively, the number itself can be divided by 3 and 9. The number 79123 is not divisible by either 3 or 9, so how the sum of its digits (22) cannot be divided by these numbers.
A characteristic feature of dividing by 4, 8, 16 and so on.
The figure can be divided without remainder by four, if its last two digits are zeros or are a number that can be divided by 4. In all other options, division without a remainder is not possible.
For example:
The number 75300 is divisible by 4 since the last two digits are zeros; 48834 is not divisible by 4, since the last two digits give the number 34, which is not divisible by 4; 35908 is divisible by 4 because the last two digits of 08 give the number 8, which is divisible by 4.
A similar principle is suitable for the test of divisibility by eight. A number is divisible by eight if its last three digits are zeros or form a number divisible by 8. In other cases, the quotient obtained from division will not be a whole number.
The same properties for division by 16, 32, 64 etc., but they are not used in everyday calculations.
A characteristic feature of divisibility by 6.
The number is divisible by six, if it is divisible by both two and three, with all other options, division without a remainder is impossible.
For example:
126 is divisible by 6 because it is divisible by both 2 (the final even number is 6) and 3 (the sum of the digits 1 + 2 + 6 = 9 is divisible by three)
A characteristic feature of divisibility by 7.
The number is divisible by seven if the difference between its doubled last number and the “number left without the last digit” is divisible by seven, then the number itself is divisible by seven.
For example:
The number is 296492. Take the last digit “2”, double it, it comes out to 4. Subtract 29649 - 4 = 29645. It is problematic to find out whether it is divisible by 7, therefore analyzed again. Next, we double the last digit “5”, the result is 10. Subtract 2964 - 10 = 2954. The result is the same, it is not clear whether it is divisible by 7, therefore we continue the analysis. We analyze with the last digit “4”, double it, it comes out 8. Subtract 295 - 8 = 287. We check two hundred eighty-seven - it is not divisible by 7, therefore we continue the search. By analogy, we double the last digit “7”, it becomes 14. Subtract 28 - 14 = 14. The number 14 is divided by 7, so the original number is divided by 7.
Characteristic feature of divisibility by 11.
On eleven Only those numbers are divided in which the result of adding the digits located in odd places is either equal to the sum of the digits located in even places or is different from a number divisible by eleven.
For example:
The number 103,785 is divisible by 11, since the sum of the digits in odd places, 1 + 3 + 8 = 12, is equal to the sum of the digits in even places, 0 + 7 + 5 = 12. The number 9,163,627 is divisible by 11, since the sum of digits placed in odd places is 9 + 6 + 6 + 7 = 28, and the sum of digits placed in even places is 1 + 3 + 2 = 6; the difference between the numbers 28 and 6 is 22, and this number is divisible by 11. The number 461,025 is not divisible by 11, since the numbers 4 + 1 + 2 = 7 and 6 + 0 + 5 = 11 are not equal to each other, but their difference 11 - 7 = 4 is not divisible by 11.
Characteristic feature of divisibility by 25.
On twenty five numbers whose final two digits are zeros or form a number that can be divided by twenty-five (that is, numbers ending in 00, 25, 50, or 75) will be divided. In other cases, the number cannot be divided entirely by 25.
For example:
9450 is divisible by 25 (ending in 50); 5085 is not divisible by 25.
Divisibility test
Sign of divisibility- a rule that allows you to relatively quickly determine whether a number is a multiple of a predetermined number without having to do the actual division. As a rule, it is based on actions with part of the digits from the number written in the positional number system (usually decimal).
There are several simple rules that allow you to find small divisors of a number in the decimal number system:
Test for divisibility by 2
Test for divisibility by 3
Test for divisibility by 4
Divisibility test by 5
Test for divisibility by 6
Test for divisibility by 7
Divisibility test by 8
Divisibility test by 9
Divisibility test by 10
Divisibility test by 11
Divisibility test by 12
Divisibility test by 13
Divisibility test by 14
Divisibility test by 15
Divisibility test by 17
Divisibility test by 19
Test for divisibility by 23
Test for divisibility by 25
Divisibility test by 99
Let's divide the number into groups of 2 digits from right to left (the leftmost group can have one digit) and find the sum of these groups, considering them two-digit numbers. This sum is divisible by 99 if and only if the number itself is divisible by 99.
Divisibility test by 101
Let's divide the number into groups of 2 digits from right to left (the leftmost group can have one digit) and find the sum of these groups with alternating signs, considering them two-digit numbers. This sum is divisible by 101 if and only if the number itself is divisible by 101. For example, 590547 is divisible by 101, since 59-05+47=101 is divisible by 101).
Test for divisibility by 2 n
A number is divisible by the nth power of two if and only if the number formed by its last n digits is divisible by the same power.
Divisibility test by 5 n
A number is divisible by the nth power of five if and only if the number formed by its last n digits is divisible by the same power.
Divisibility test by 10 n − 1
Let's divide the number into groups of n digits from right to left (the leftmost group can have from 1 to n digits) and find the sum of these groups, considering them n-digit numbers. This amount is divided by 10 n− 1 if and only if the number itself is divisible by 10 n − 1 .
Divisibility test by 10 n
A number is divisible by the nth power of ten if and only if its last n digits are
Loading...