What is divisible by 12 and 13. Creative work "signs of divisibility"

Test for divisibility by 3
A number is divisible by 3 if and only if the sum of its digits is divisible by 3.

Test for divisibility by 4
A number is divisible by 4 if and only if the last two digits of the number are zeros or divisible by 4.

Divisibility test by 5
A number is divisible by 5 if and only if the last digit is divisible by 5 (that is, equal to 0 or 5).

Divisibility test by 6
A number is divisible by 6 if and only if it is divisible by 2 and 3.

Test for divisibility by 7
A number is divisible by 7 if and only if the result of subtracting twice the last digit from that number without the last digit is divisible by 7 (for example, 259 is divisible by 7, since 25 - (2 9) = 7 is divisible by 7).

Divisibility test by 8
A number is divisible by 8 if and only if its last three digits are zeros or form a number that is divisible by 8.

Divisibility test by 9
A number is divisible by 9 if and only if the sum of its digits is divisible by 9.

Divisibility test by 10
A number is divisible by 10 if and only if it ends in zero.

Divisibility test by 11
A number is divisible by 11 if and only if the sum of the digits with alternating signs is divisible by 11 (that is, 182919 is divisible by 11, since 1 - 8 + 2 - 9 + 1 - 9 = -22 is divisible by 11) - a consequence of the fact that that all numbers of the form 10 n when divided by 11 leave a remainder of (-1) n .

Divisibility test by 12
A number is divisible by 12 if and only if it is divisible by 3 and 4.

Divisibility test by 13
A number is divisible by 13 if and only if the number of its tens added to four times the number of ones is a multiple of 13 (for example, 845 is divisible by 13, since 84 + (4 5) = 104 is divisible by 13).

Divisibility test by 14
A number is divisible by 14 if and only if it is divisible by 2 and 7.

Divisibility test by 15
A number is divisible by 15 if and only if it is divisible by 3 and 5.

Divisibility test by 17
A number is divisible by 17 if and only if the number of its tens, added with 12 times the number of units, is a multiple of 17 (for example, 29053→2905+36=2941→294+12=306→30+72=102→10+ 24 = 34. Since 34 is divisible by 17, then 29053 is divisible by 17). The sign is not always convenient, but it has a certain meaning in mathematics. There is a slightly simpler way - A number is divisible by 17 if and only if the difference between the number of its tens and five times the number of units is a multiple of 17 (for example, 32952→3295-10=3285→328-25=303→30-15=15. since 15 is not divisible by 17, then 32952 is not divisible by 17)

Divisibility test by 19
A number is divisible by 19 if and only if the number of its tens added to twice the number of ones is a multiple of 19 (for example, 646 is divisible by 19, since 64 + (6 2) = 76 is divisible by 19).

Test for divisibility by 23
A number is divisible by 23 if and only if its hundreds number added to triple its tens number is a multiple of 23 (for example, 28842 is divisible by 23, since 288 + (3 * 42) = 414 continues 4 + (3 * 14) = 46 is obviously divisible by 23).

Test for divisibility by 25
A number is divisible by 25 if and only if its last two digits are divisible by 25 (that is, forming 00, 25, 50 or 75) or the number is a multiple of 5.

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.

Test for divisibility 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).

Signs of divisibility of numbers- these are rules that allow you to relatively quickly find out, without dividing, whether this number is divisible by a given number without a remainder.
Some of signs of divisibility quite simple, some more complicated. On this page you will find both signs of divisibility prime numbers, such as, for example, 2, 3, 5, 7, 11, and signs of divisibility of composite numbers, such as 6 or 12.
Hope, this information will be useful to you.
Happy learning!

Test for divisibility by 2

This is one of the simplest signs of divisibility. It sounds like this: if the notation of a natural number ends with an even digit, then it is even (divisible without a remainder by 2), and if the notation of a natural number ends with an odd digit, then this number is odd.
In other words, if the last digit of a number is 2 , 4 , 6 , 8 or 0 - the number is divisible by 2, if not, then it is not divisible
For example, numbers: 23 4 , 8270 , 1276 , 9038 , 502 are divisible by 2 because they are even.
A numbers: 23 5 , 137 , 2303
They are not divisible by 2 because they are odd.

Test for divisibility by 3

This sign of divisibility has completely different rules: if the sum of the digits of a number is divisible by 3, then the number is divisible by 3; If the sum of the digits of a number is not divisible by 3, then the number is not divisible by 3.
This means that in order to understand whether a number is divisible by 3, you just need to add together the numbers that make it up.
It looks like this: 3987 and 141 are divisible by 3, because in the first case 3+9+8+7= 27 (27:3=9 - divisible by 3), and in the second 1+4+1= 6 (6:3=2 - also divisible by 3).
But the numbers: 235 and 566 are not divisible by 3, because 2+3+5= 10 and 5+6+6= 17 (and we know that neither 10 nor 17 are divisible by 3 without a remainder).

Test for divisibility by 4

This sign of divisibility will be more complicated. If the last 2 digits of a number form a number divisible by 4 or it is 00, then the number is divisible by 4, otherwise the given number is not divisible by 4 without a remainder.
For example: 1 00 and 3 64 are divisible by 4 because in the first case the number ends in 00 , and in the second on 64 , which in turn is divisible by 4 without a remainder (64:4=16)
Numbers 3 57 and 8 86 are not divisible by 4 because neither 57 neither 86 are not divisible by 4, which means they do not correspond to this criterion of divisibility.

Divisibility test by 5

And again we have a fairly simple sign of divisibility: if the notation of a natural number ends with the number 0 or 5, then this number is divisible by 5 without a remainder. If the notation of a number ends with another digit, then the number is not divisible by 5 without a remainder.
This means that any numbers ending in digits 0 And 5 , for example 1235 5 and 43 0 , fall under the rule and are divisible by 5.
And, for example, 1549 3 and 56 4 do not end with the number 5 or 0, which means they cannot be divided by 5 without a remainder.

Divisibility test by 6

Before us composite number 6, which is the product of the numbers 2 and 3. Therefore, the sign of divisibility by 6 is also composite: in order for a number to be divisible by 6, it must correspond to two signs of divisibility at the same time: the sign of divisibility by 2 and the sign of divisibility by 3. Please note , that such a composite number as 4 has an individual sign of divisibility, because it is the product of the number 2 by itself. But let's return to the test of divisibility by 6.
The numbers 138 and 474 are even and meet the criteria for divisibility by 3 (1+3+8=12, 12:3=4 and 4+7+4=15, 15:3=5), which means they are divisible by 6. But 123 and 447, although they are divisible by 3 (1+2+3=6, 6:3=2 and 4+4+7=15, 15:3=5), but they are odd, which means they do not correspond to the criterion of divisibility by 2, and therefore do not correspond to the criterion of divisibility by 6.

Test for divisibility by 7

This test of divisibility is more complex: a number is divisible by 7 if the result of subtracting twice the last digit from the number of tens of this number is divisible by 7 or equal to 0.
It sounds quite confusing, but in practice it is simple. See for yourself: the number 95 9 is divisible by 7 because 95 -2*9=95-18=77, 77:7=11 (77 is divided by 7 without a remainder). Moreover, if difficulties arise with the number obtained during the transformation (due to its size it is difficult to understand whether it is divisible by 7 or not, then this procedure can be continued as many times as you deem necessary).
For example, 45 5 and 4580 1 have the properties of divisibility by 7. In the first case, everything is quite simple: 45 -2*5=45-10=35, 35:7=5. In the second case we will do this: 4580 -2*1=4580-2=4578. It is difficult for us to understand whether 457 8 by 7, so let's repeat the process: 457 -2*8=457-16=441. And again we will use the sign of divisibility, since we still have before us three digit number 44 1. So, 44 -2*1=44-2=42, 42:7=6, i.e. 42 is divisible by 7 without a remainder, which means 45801 is divisible by 7.
Here are the numbers 11 1 and 34 5 is not divisible by 7 because 11 -2*1=11-2=9 (9 is not divisible by 7) and 34 -2*5=34-10=24 (24 is not divisible by 7 without a remainder).

Divisibility test by 8

The test for divisibility by 8 sounds like this: if the last 3 digits form a number divisible by 8, or it is 000, then the given number is divisible by 8.
Numbers 1 000 or 1 088 divisible by 8: the first one ends in 000 , the second 88 :8=11 (divisible by 8 without remainder).
And here are the numbers 1 100 or 4 757 are not divisible by 8 because numbers 100 And 757 are not divisible by 8 without a remainder.

Divisibility test by 9

This sign of divisibility is similar to the sign of divisibility by 3: if the sum of the digits of a number is divisible by 9, then the number is divisible by 9; If the sum of the digits of a number is not divisible by 9, then the number is not divisible by 9.
For example: 3987 and 144 are divisible by 9, because in the first case 3+9+8+7= 27 (27:9=3 - divisible by 9 without remainder), and in the second 1+4+4= 9 (9:9=1 - also divisible by 9).
But the numbers: 235 and 141 are not divisible by 9, because 2+3+5= 10 and 1+4+1= 6 (and we know that neither 10 nor 6 are divisible by 9 without a remainder).

Signs of divisibility by 10, 100, 1000 and other digit units

I combined these signs of divisibility because they can be described in the same way: a number is divided by a digit unit if the number of zeros at the end of the number is greater than or equal to the number of zeros at a given digit unit.
In other words, for example, we have the following numbers: 654 0 , 46400 , 867000 , 6450 . of which all are divisible by 1 0 ; 46400 and 867 000 are also divisible by 1 00 ; and only one of them is 867 000 divisible by 1 000 .
Any numbers that have less trailing zeroes than the digit unit are not divisible by that digit unit, for example 600 30 and 7 93 not divisible 1 00 .

Divisibility test by 11

In order to find out whether a number is divisible by 11, you need to obtain the difference between the sums of the even and odd digits of this number. If this difference is equal to 0 or is divisible by 11 without a remainder, then the number itself is divisible by 11 without a remainder.
To make it clearer, I suggest looking at examples: 2 35 4 is divisible by 11 because ( 2 +5 )-(3+4)=7-7=0. 29 19 4 is also divisible by 11, since ( 9 +9 )-(2+1+4)=18-7=11.
Here's 1 1 1 or 4 35 4 is not divisible by 11, since in the first case we get (1+1)- 1 =1, and in the second ( 4 +5 )-(3+4)=9-7=2.

Divisibility test by 12

The number 12 is composite. Its sign of divisibility is compliance with the signs of divisibility by 3 and 4 at the same time.
For example, 300 and 636 correspond to both the signs of divisibility by 4 (the last 2 digits are zeros or are divisible by 4) and the signs of divisibility by 3 (the sum of the digits of both the first and third numbers are divisible by 3), but finally, they are divisible by 12 without a remainder.
But 200 or 630 are not divisible by 12, because in the first case the number meets only the criterion of divisibility by 4, and in the second - only the criterion of divisibility by 3. but not both criteria at the same time.

Divisibility test by 13

A sign of divisibility by 13 is that if the number of tens of a number added to the units of this number multiplied by 4 is a multiple of 13 or equal to 0, then the number itself is divisible by 13.
Let's take for example 70 2. So, 70 +4*2=78, 78:13=6 (78 is divisible by 13 without a remainder), which means 70 2 is divisible by 13 without a remainder. Another example is a number 114 4. 114 +4*4=130, 130:13=10. The number 130 is divisible by 13 without a remainder, which means the given number corresponds to the criterion of divisibility by 13.
If we take the numbers 12 5 or 21 2, then we get 12 +4*5=32 and 21 +4*2=29, respectively, and neither 32 nor 29 are divisible by 13 without a remainder, and therefore given numbers are not divisible by 13 without a remainder.

Divisibility of numbers

As can be seen from the above, it can be assumed that to any of natural numbers you can choose your own individual sign of divisibility or a “composite” sign if the number is a multiple of several different numbers. But as practice shows, mainly than larger number, the more complex its sign. It is possible that the time spent checking the divisibility criterion may be equal to or greater than the division itself. That's why we usually use the simplest signs of divisibility.

CHISTENSKY UVK "COMMON EDUCATION SCHOOL"

I – III STAGES - GYMNASIUM"

DIRECTION MATHEMATICS

"SIGNS OF DIVISION"

I've done the work

7a grade student

Zhuravlev David

Scientific director

specialist of the highest category

Fedorenko Irina Vitalievna

Clean, 2013

Table of contents

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1. Divisibility of numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1 Signs of divisibility by 2, 5, 10, 3 and 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Signs of divisibility by 4, 25 and 50. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Divisibility criteria by 8 and 125. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Simplification of the test for divisibility by 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Signs of divisibility by 6, 12, 15, 18, 45, etc. . . . . . . . . . . . . . . . . . . . . . . 6

1. Test for divisibility by 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2. Simple signs divisibility by prime numbers. . . . . . . . . . . . . . . . . 7

2.1 Tests for divisibility by 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Tests for divisibility by 11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Tests for divisibility by 13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Tests for divisibility by 19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3. Combined sign of divisibility by 7, 11 and 13. . . . . . . . . . . . . . . . . . 9

4. Old and new about divisibility by 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5. Extension of the divisibility test by 7 to other numbers. . . . . . 12

6. Generalized test of divisibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7. The curiosity of divisibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

INTRODUCTION

If you want to learn how to swim, then boldly enter the water, and if you want to learn how to solve problems, then solve them.

D. Polya

There are many branches in mathematics and one of them is divisibility of numbers.

Mathematicians of past centuries have come up with many convenient tricks to facilitate the calculations and calculations that abound in solving mathematical problems. A completely reasonable way out of the situation, because they had neither calculators nor computers. In some situations, the ability to use convenient calculation methods greatly facilitates solving problems and significantly reduces the time spent on them.

Such useful methods of calculation undoubtedly include signs of divisibility by a number. Some of them are easier - these signs of divisibility of numbers by 2, 3, 5, 9, 10 are studied as part of a school course, and some are quite complex and are of more research interest than practical interest. However, it is always interesting to check each of the divisibility tests on specific numbers.

Goal of the work: expand your understanding of the properties of numbers related to divisibility;

Tasks:

Get acquainted with various signs of divisibility of numbers;

Systematize them;

Develop skills in applying introduced rules and algorithms to establish the divisibility of numbers.

Divisibility of numbers

The test of divisibility is a rule by which, without performing division, you can determine whether one number is divisible by another.

Divisibility of a sum. If each term is divisible by a number, then the sum is divisible by that number.

Example 1.1

32 is divisible by 4, 16 is divisible by 4, so the sum of 32 + 16 is divisible by 4.

Divisibility of difference. If the minuend and the subtrahend are divisible by some number, then the difference is divided by this number.

Example 1.2

777 is divisible by 7, 49 is divisible by 7, which means the difference 777 – 49 is divisible by 7.

Divisibility of a product by a number. If in a product at least one of the factors is divisible by a certain number, then the product is also divisible by this number.

Example 1.3

15 is divisible by 3, which means the product 15∙17∙23 is divisible by 3.

Divisibility of a number by a product. If a number is divisible by a product, then it is divided by each of the factors of that product.

Example 1.4

90 is divisible by 30, 30 = 2∙3∙5, which means 30 is divisible by 2, 3, and 5.

B. Pascal made a great contribution to the study of divisibility criteria for numbers.Blaise Pascal (Blaise Pascal) (1623–1662), French religious thinker, mathematician and physicist, one of greatest minds 17th century.He formulated the following criterion of divisibility, which bears his name:

Natural number A will be divided by another natural number b only if the sum of the products of the digits of the number A into the corresponding remainders obtained by dividing the digit units by the number b , is divided by this number.

1.1 Tests for divisibility by 2, 5, 10, 3 and 9

At school we study the signs of divisibility by 2, 3, 5, 9, 10.

Test for divisibility by 10. All those and only those numbers whose writing ends with the number 0 are divisible by 10.

Test for divisibility by 5. All those and only those numbers whose notation ends with the number 0 or 5 are divisible by 5.

Test for divisibility by 2. All those and only those numbers whose writing ends with an even digit are divisible by 2: 2,4,6,8 or 0.

Test for divisibility by 3 and 9. 3 and 9 are divisible by all those and only those numbers whose sum of digits is divisible by 3 or 9, respectively.

By writing a number (by its last digits), you can also establish the divisibility of the number by 4, 25, 50, 8 and 125.

1.2 Divisibility criteria by 4, 25 and 50

Divide by 4, 25 or 50 are those and only those numbers that end with two zeros or whose last two digits express a number divisible by 4, 25 or 50, respectively.

Example 1.2.1

The number 97300 ends with two zeros, which means it is divisible by 4, 25, and 50.

Example 1.2.2

The number 81764 is divisible by 4 because the number formed by the last two digits, 64, is divisible by 4.

Example 1.2.3

The number 79,450 is divisible by 25 and 50, since the number formed by the last two digits 50 is divisible by both 25 and 50.

1.3 Tests for divisibility by 8 and 125

Divisible by 8 or 125 are those and only those numbers that end with three zeros or whose last three digits express a number divisible by 8 or 125, respectively.

Example 1.3.1

The number 853,000 ends with three zeros, which means it is divisible by both 8 and 125.

Example 1.3.2

The number 381,864 is divisible by 8 because the number formed by the last three digits of 864 is divisible by 8.

Example 1.3.3

The number 179250 is divisible by 125, since the number formed by the last three digits of 250 is divisible by 125.

1.4 Simplification of the test for divisibility by 8

The question of the divisibility of a certain number reduces to the question of the divisibility by 8 of a certain three-digit number, butat the same time, nothing is said about how, in turn, to quickly find out whether this three-digit number is divisible by 8. The divisibility of a three-digit number by 8 is also not always immediately visible; you actually have to carry out the division.

The question naturally arises: is it possible to simplify the test for divisibility by 8? It is possible if you supplement it with a special sign of divisibility of a three-digit number by 8.

Any three-digit number is divisible by 8. two-digit number, formed by the hundreds and tens digits, added with half the units number, is divisible by 4.

Example 1.4.1

Is the number 592 divisible by 8?

Solution.

We separate 592 units from the number and add half of their number to the number of the next two digits (tens and hundreds).

We get: 59 + 1 = 60.

The number 60 is divisible by 4, which means the number 592 is divisible by 8.

Answer: shares.

1.5 Signs of divisibility by 6, 12, 15, 18, 45, etc.

Using the property of divisibility of a number by a product, from the above-mentioned signs of divisibility we obtain signs of divisibility by 6, 12, 15, 18, 24, etc.

Test for divisibility by 6. Those and only those numbers that are divisible by 2 and 3 are divisible by 6.

Example 1.5.1

The number 31242 is divisible by 6 because it is divisible by both 2 and 3.

Test for divisibility by 12. Those and only those numbers that are divisible by 4 and 3 are divisible by 12.

Example 1.5.2

The number 316224 is divisible by 12 because it is divisible by both 4 and 3.

Test for divisibility by 15. Those and only those numbers that are divisible by 3 and 5 are divisible by 15.

Example 1.5.3

The number 812445 is divisible by 15 because it is divisible by both 3 and 5.

Test for divisibility by 18. Those and only those numbers that are divisible by 2 and 9 are divisible by 18.

Example 1.5.4

The number 817254 is divisible by 18 because it is divisible by both 2 and 9.

Test for divisibility by 45. Those and only those numbers that are divisible by 5 and 9 are divisible by 45.

Example 1.5.5

The number 231,705 is divisible by 45 because it is divisible by both 5 and 9.

There is another sign that numbers are divisible by 6.

1.6 Test for divisibility by 6

To check whether a number is divisible by 6, you need to:

The number of hundreds multiplied by 2,

Subtract the result obtained from the number after the number of hundreds.

If the result is divisible by 6, then the entire number is divisible by 6. Example 1.6.1

Is the number 138 divisible by 6?

Solution.

The number of hundreds is 1·2=2, 38-2=36, 36:6, which means 138 is divisible by 6.

Simple tests for divisibility by prime numbers

A number is called prime if it has only two divisors (one and the number itself).

2.1 Tests for divisibility by 7

To find out if a number is divisible by 7, you need to:

Numbers up to tens multiplied by two

Add the remaining number to the result.

Check whether the result is divisible by 7 or not.

Example 2.1.1

Is the number 4690 divisible by 7?

Solution.

The number to the tens is 46·2=92, 92+90=182, 182:7=26, which means 4690 is divisible by 7.

2.2 Tests for divisibility by 11

A number is divisible by 11 if the difference between the sum of the digits in odd places and the sum of the digits in even places is a multiple of 11.

The difference may be negative number or be equal to zero, but must be a multiple of 11.

Example 2.2.1

Is the number 100397 divisible by 11?

Solution.

The sum of the numbers in even places: 1+0+9=10.

The sum of the numbers in odd places: 0+3+7=10.

Difference of sums: 10 – 10=0, 0 is a multiple of 11, which means 100397 is divisible by 11.

2.3 Tests for divisibility by 13

A number is divisible by 13 if and only if the result of subtracting the last digit multiplied by 9 from that number without the last digit is divisible by 13.

Example 2.3.1

The number 858 is divisible by 13, since 85 - 9∙8 = 85 – 72 = 13 is divisible by 13.

2.4 Tests for divisibility by 19

A number is divisible by 19 without a remainder when the number of its tens added to twice the number of units is divisible by 19.

Example 2.4.1

Determine whether 1026 is divisible by 19.

Solution.

There are 102 tens and 6 ones in 1026. 102 + 2∙6 = 114;

Similarly 11 + 2∙4 = 19.

As a result of performing two successive steps, we get the number 19, which is divisible by 19, therefore, the number 1026 is divisible by 19.

Combined test for divisibility by 7, 11 and 13

In the table of prime numbers, the numbers 7, 11 and 13 are located next to each other. Their product is equal to: 7 ∙ 11 ∙ 13 = 1001 = 1000 + 1. This means that the number 1001 is divisible by 7, 11, and 13.

If any three-digit number is multiplied by 1001, then the product will be written in the same digits as the multiplicand, only repeated twice:abc– three-digit number;abc∙1001 = abcabc.

Therefore, all numbers of the form abcabc are divisible by 7, 11 and 13.

These regularities make it possible to reduce the solution to the question of the divisibility of a multi-digit number by 7 or 11 or 13 to the divisibility of some other number by them - no more than a three-digit one.

If the difference between the sums of the sides of a given number, taken one at a time, is divisible by 7 or 11 or 13, then the given number is divisible by 7 or 11 or 13, respectively.

Example 3.1

Determine whether the number 42,623,295 is divisible by 7, 11 and 13.

Solution.

Let's divide this number from right to left into edges of 3 digits. The leftmost edge may not have three digits. Let us determine which of the numbers 7, 11 or 13 divides the difference between the sums of the faces of a given number:

623 - (295 + 42) = 286.

The number 286 is divisible by 11 and 13, but it is not divisible by 7. Therefore, the number 42,623,295 is divisible by 11 and 13, but not divisible by 7.

Old and new about divisibility by 7

For some reason, the number 7 became very popular with the people and became part of their songs and sayings:

Try it on seven times, cut it once.

Seven troubles, one answer.

Seven Fridays a week.

One with a bipod, and seven with a spoon.

Too many cooks spoil the broth.

The number 7 is rich not only in sayings, but also in various signs of divisibility. You already know two signs of divisibility by 7 (in combination with other numbers). There are also several individual signs of divisibility by 7.

We will explain the first sign of divisibility by 7 with an example.

Let's take the number 5236. Let's write this number as follows:

5 236 = 5∙10 3 + 2∙10 2 + 3∙10 + 6

and everywhere we replace base 10 with base 3: 5∙3 3 + 2∙3 2 + 3∙3 + 6 = 168

If the resulting number is divisible (not divisible) by 7, then the given number is divisible (not divisible) by 7.

Since 168 is divisible by 7, then 5236 is divisible by 7.

Modification of the first sign of divisibility by 7. Multiply the first digit on the left of the test number by 3 and add next digit; multiply the result by 3 and add the next digit, etc. until the last digit. To simplify, after each action it is allowed to subtract 7 or a multiple of seven from the result. If the final result is divisible (not divisible) by 7, then this number is divisible (not divisible) by 7. For the previously selected number 5,236:

5∙3 = 15; (15 – 14 = 1); 1 + 2 = 3; 3∙3 = 9; (9 – 7 = 2); 2 + 3 = 5; 5∙3 = 15; (15 – 14 = 1); 1 + 6 = 7 – divisible by 7, which means 5,236 is divisible by 7.

The advantage of this rule is that it is easy to apply mentally.

The second sign of divisibility by 7. In this sign you must act in exactly the same way as in the previous one, with the only difference that multiplication should begin not with the leftmost digit of a given number, but with the rightmost one and multiply not by 3, but by 5 .

Example 4.1

Is 37,184 divisible by 7?

Solution.

4∙5=20; (20 - 14 = 6); 6 + 8=14; (14 - 14 = 0); 0∙5 = 0; 0+1= 1; 1∙5 = 5; adding the number 7 can be skipped, since the number 7 is subtracted from the result obtained; 5∙5 = 25; (25 - 21= 4); 4 + 3 = 7 – divisible by 7, which means the number 37,184 is divisible by 7.

The third sign of divisibility by 7. This sign is less easy to implement in the mind, but it is also very interesting.

Double the last digit and subtract the second from the right, double the result and add the third from the right, etc., alternating subtraction and addition and decreasing each result, where possible, by 7 or a multiple of seven. If the final result is divisible (not divisible) by 7, then the tested number is divisible (not divisible) by 7.

Example 4.2

Is 889 divisible by 7?

Solution.

9∙2 = 18; 18 – 8 = 10; 10∙2 = 20; 20 + 8 = 28 or

9∙2 = 18; (18 – 7 = 11) 11 – 8 = 3; 3∙2 = 6; 6 + 8 = 14 – divisible by 7, which means the number 889 is divisible by 7.

And more signs of divisibility by 7. If any two-digit number is divisible by 7, then the inverse number increased by the tens digit of the given number is also divisible by 7.

Example 4.3

14 is divisible by 7, therefore divisible by 7 and the number 41 + 1.

35 is divisible by 7, therefore 53 + 3 is divisible by 7.

If any three-digit number is divisible by 7, then the inverted number is divided by 7, reduced by the difference between the units and hundreds digits of the given number.

Example 4.4

The number 126 is divisible by 7. Therefore, the number 621 - (6 - 1) is divisible by 7, that is, 616.

Example 4.5

The number 693 is divisible by 7. Therefore, the number 396 is also divisible by 7 - (3 - 6), that is, 399.

Extension of the divisibility test by 7 to other numbers

The above three criteria for the divisibility of numbers by 7 can be used to determine the divisibility of a number by 13, 17 and 19.

To determine whether a given number is divisible by 13, 17 or 19, you must multiply the leftmost digit of the test number by 3, 7 or 9, respectively, and subtract the next digit; multiply the result again by 3, 7 or 9, respectively, and add the next digit, etc., alternating between subtracting and adding subsequent digits after each multiplication. After each action, the result can be reduced or increased, respectively, by the number 13, 17, 19 or a multiple of it.

If the final result is divisible (not divisible) by 13, 17 and 19, then the given number is divisible (not divisible).

Example 5.1

Is the number 2,075,427 divisible by 19?

Solution.

2∙9=18; 18 – 0 = 18; 18∙9 = 162; (162 - 19∙8 = 162 = 10); 10 + 7 = 17; 17∙9 = 153; (153 - 19∙7 = 20); 20 – 5 = 15; 15∙9 = 135; (135 - 19∙7 = 2);

2 + 4 = 6; 6∙9 = 54; (54 - 19∙2 = 16); 16 – 2 = 14; 14∙9 = 126; (126 - 19∙6 = = 12); 12 + 7 = 19 – divisible by 19, which means 2,075,427 is divisible by 19.

Generalized test of divisibility

The idea of ​​dissecting a number on its face and then adding them to determine the divisibility of a given number turned out to be very fruitful and led to a uniform test for the divisibility of multi-digit numbers into a fairly large group of prime numbers. One of the groups of “lucky” divisors are all integer factors p of the number d = 10n + 1, where n = 1, 2, 3,4, ... (with large values n the practical meaning of the attribute is lost).

101

101

1001

7, 11, 13

10001

73, 137

2) fold the edges one at a time, starting from the far right;

3) fold the remaining faces;

4) from larger amount subtract the smaller one.

If the result is divisible by p, then the given number is divisible by p.

So, to determine the divisibility of a number by 11 (p = 11), we cut the number on the face along one digit (n = 1). Proceeding further as indicated, we arrive at the well-known test for divisibility by 11.

When determining whether a number is divisible by 7, 11 or 13 (p = 7, 11, 13), we cut off 3 digits (n = 3). When determining whether a number is divisible by 73 and 137, we cut off 4 digits each (n = 4).

Example 6.1

Find out the divisibility of the fifteen-digit number 837,362,172,504,831 by 73 and 137 (p = 73, 137, n = 4).

Solution.

We break the number down: 837 3621 7250 4831.

Add the faces one at a time: 4931 + 3621 = 8452; 7250 + 837 = 8087.

Subtract the smaller amount from the larger amount: 8452-8087 = 365.

365 is divisible by 73, but not divisible by 137; This means that the given number is divisible by 73, but not divisible by 137.

The second group of “lucky” divisors are all integer factors p of the number d = 10n -1, where n = 1, 3, 5, 7,…

The number d = 10n -1 gives the following divisors:

n

d

p

1

9

3

3

999

37

5

99 999

41, 271

To determine whether a number is divisible by any of these numbers p, you need to:

1) cut the given number from right to left (from ones) into n-digit edges (each p has its own n; the leftmost edge can have less than n digits);

2) add all the edges.

If the result obtained is divisible (not divisible) by p, then the given number is divisible (not divisible).

Note that 999 = 9∙111, which means that 111 is divisible by 37, but then the numbers 222, 333, 444, 555, 666, 777 and 888 are also divisible by 37.

Similarly: 11,111 is divisible by 41 and 271.

The curiosity of divisibility

In conclusion, I would like to present four amazing ten-digit numbers:

2 438 195 760; 4 753 869 120;3 785 942 160; 4 876 391 520.

Each of them has all the digits from 0 to 9, but each digit only once and each of these numbers is divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 , 15, 16, 17 and 18.

conclusions

As a result of this work, I have expandedknowledge in mathematics. II learned that in addition to the signs of divisibility by 2, 3, 5, 9 and 10 that I know, there are also signs of divisibility by 4, 6, 7, 8, 11, 12, 13, 14, 15, 19, 25, 50, 125 and other numbers , and the signs of divisibility by the same number can be different, which means there is always room for creativity.

The work is theoretical in nature andpractical use. This research will be useful in preparing for Olympiads and competitions.

Having become acquainted with the signs of divisibility of numbers, I believe that I can use the knowledge gained in my educational activities, independently apply one or another feature to a specific task, apply the learned features in a real situation. In the future, I plan to continue working on studying the signs of divisibility of numbers.

Literature

1. N. N. Vorobyov “Signs of divisibility” Moscow “Science” 1988

2. K. I. Shchevtsov, G. P. Bevz “Handbook of elementary mathematics” Kyiv “Naukova Duma” 1965

3. M. Ya. Vygodsky “Handbook of Elementary Mathematics” Moscow “Science” 1986

4. Internet resources

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, which 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.