The use of divisibility skills simplifies calculations, and proportionately increases the speed of their execution. Let us examine in detail the main characteristic divisibility features.
The most straightforward divisibility criterion for units: All numbers are divisible by one. It is also elementary with the criteria for divisibility by two, five, ten... You can divide even numbers by two, or the one with the final digit 0, by five - the number with the final digit 5 or 0. Only those numbers with the final digit 0 will be divided by ten, by 100 - only those numbers with two trailing zeros, on 1000 - only those with three trailing zeros.
For example:
The number 79516 can be divided by 2, since it ends with 6 - an even number; 9651 cannot be divided by 2, since 1 is an odd digit; 1790 will be divided by 2 since the trailing digit is zero. 3470 is divided by 5 (the final digit is 0); 1054 is not divisible by 5 (ending number 4). 7800 is divisible by 10 and by 100; 542000 is divisible by 10, 100, 1000.
Less widely known, but very easy to use are characteristic divisibility features on the 3 and 9 , 4 , 6 and 8, 25 ... There are also characteristic features of divisibility by 7, 11, 13, 17, 19 and so on, but they are used much less frequently in practice.
The salient feature of division by 3 and 9.
On the three and / or on nine those numbers for which the result of adding the 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 divided by 3 and divided by 9, respectively, and the number itself can be divided by 3 and 9. The number 79123 cannot be divided by either 3 or 9, so as the sum of its digits (22) is not divisible by these numbers.
The characteristic feature of division by 4, 8, 16 and so on.
The figure can be completely divided by four, if it has two last digits zeros or is a number that can be divided by 4. In all other variants, division without a remainder is not possible.
For example:
The number 75300 is divided by 4, since the last two digits are zeros; 48834 is not divisible by 4, since the last two digits give 34, not divisible by 4; 35908 is divisible by 4 because the last two digits 08 give 8 divisible by 4.
A similar principle is applicable for the divisibility criterion by eight... A number is divisible by eight if its last three digits are zeros or form a number divisible by 8. Otherwise, the quotient obtained from division will not be an integer.
The same properties for division by 16, 32, 64 and so on, but they are not used in everyday computing.
The salient feature of divisibility by 6.
The number is divided 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, since 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)
The salient feature of divisibility by 7.
The number is divided 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:
Number 296492. Take the last digit "2", double it, it turns out 4. Subtract 29649 - 4 = 29645. It is problematic to find out whether it is divisible by 7, therefore analyzed again. Then we double the last digit "5", it turns out 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 it with the last digit "4", double it, it turns out 8. Subtract 295 - 8 = 287. We compare two hundred and eighty-seven - it is not divisible by 7, in this regard, we continue the search. By analogy, we double the last digit "7", it turns out 14. Subtract 28 - 14 = 14. The number 14 is divided by 7, so the original number is divided by 7.
The salient feature of divisibility by 11.
On the eleven only those numbers are divided in which the result of the addition of the digits located in odd places is either equal to the sum of the digits located in even places, or differs by a number divisible by eleven.
For example:
The number 103 785 is divisible by 11, since the sum of the numbers in the odd places, 1 + 3 + 8 = 12, is equal to the sum of the numbers in the even places, 0 + 7 + 5 = 12. The number 9 163 627 is divided by 11, because the sum of the digits in the odd places is 9 + 6 + 6 + 7 = 28, and the sum of the digits in the 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.
The salient feature of divisibility by 25.
On the twenty five will divide numbers whose trailing two digits are zeros or make up a number that can be divided by twenty-five (that is, numbers ending in 00, 25, 50, or 75). With other options, the number cannot be divided entirely by 25.
For example:
9450 is divisible by 25 (ends in 50); 5085 is not a multiple of 25.
Divisibility criterion
A 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 perform the actual division. As a rule, it is based on actions with a part of digits from a number recording in a positional number system (usually decimal).
There are several simple rules for finding the small divisors of a decimal number:
Divisibility by 2
Divisibility by 3
Divisibility by 4
Divisibility by 5
Divisibility by 6
Divisibility by 7
Divisibility by 8
Divisibility by 9
Divisibility by 10
Divisibility by 11
Divisibility by 12
Divisibility by 13
Divisibility by 14
Divisibility by 15
Divisibility by 17
Divisibility by 19
Divisibility by 23
Divisibility by 25
Divisibility by 99
Divide the number into groups of 2 digits from right to left (there can be one digit in the leftmost group) and find the sum of these groups, counting them as two-digit numbers. This sum is divisible by 99 if and only if the number itself is divisible by 99.
Divisibility by 101
Divide the number into groups of 2 digits from right to left (there can be one digit in the leftmost group) 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).
Divisibility by 2 n
A number is divisible by the n-th power of two if and only if the number formed by its last n digits is divisible by the same power.
Divisibility by 5 n
A number is divisible by the n-th power of five if and only if the number formed by its last n digits is divisible by the same power.
Divisibility by 10 n − 1
Divide the number into groups of n digits from right to left (in the leftmost group there can be from 1 to n digits) and find the sum of these groups, considering them n-digit numbers. This amount is divisible by 10 n- 1 if and only if the number itself is divisible by 10 n − 1 .
Divisibility by 10 n
A number is divisible by the n-th power of ten if and only if n of its last digits is
CHISTENSKY UVK "GENERAL EDUCATIONAL SCHOOL
I – III STEPS - COLLEGE "
DIRECTION MATHEMATICS
"SIGNS OF DIVISIBILITY"
I've done the work
7a grade student
Zhuravlev David
supervisor
specialist of the highest category
Fedorenko Irina Vitalievna
Clean, 2013
Table of contents
Introduction. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 2
1. Divisibility of numbers. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 3
1.1 Divisibility criteria by 2, 5, 10, 3 and 9. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 4
1.2 Divisibility criteria by 4, 25 and 50. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 4
1.3 Divisibility criteria by 8 and 125. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... five
1.4 Simplification of the divisibility criterion by 8. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... five
1.5 Divisibility criteria by 6, 12, 15, 18, 45, etc. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 6
Divisibility by 6. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 7
2. Simple tests for divisibility by prime numbers. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 7
2.1 Divisibility by 7. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 7
2.2 Divisibility criteria by 11. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... eight
2.3 Divisibility criteria by 13. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... eight
2.4 Divisibility criteria by 19. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... nine
3.The combined criterion of divisibility by 7, 11 and 13. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... nine
4. Old and new about divisibility by 7.. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 10
5. Extension of the divisibility criterion by 7 to other numbers. ... ... ... ... ... 12
6. Generalized divisibility criterion. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 13
7. Curiosity of divisibility. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... fifteen
Conclusions. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... sixteen
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. Poya
There are many sections in mathematics, and one of them is the divisibility of numbers.
The mathematicians of past centuries have come up with many handy tricks to make the calculations and calculations that are rife with the solution of mathematical problems. Quite a reasonable way out of the situation, because they did not have calculators or computers. In some situations, the ability to use convenient methods of calculation greatly facilitates the solution of problems and significantly reduces the time spent on them.
Undoubtedly, criteria for divisibility by a number are among such useful calculation techniques. Some of them are easier - these signs of divisibility of numbers by 2, 3, 5, 9, 10 are studied within the framework of the school course, and some are quite complex and are more of research interest than practical. However, it is always interesting to check each of the divisibility criteria for specific numbers.
Purpose of work: to expand the understanding of the properties of numbers related to divisibility;
Tasks:
Get acquainted with various signs of divisibility of numbers;
Systematize them;
To form the skills of applying the introduced rules, algorithms for establishing the divisibility of numbers.
Divisibility of numbers
Divisibility is a rule by which, without performing division, you can determine whether one number is divisible by another.
Divisibility of the amount. If each term is divisible by some number, then the sum is also divisible by this number.
Example 1.1
32 is divisible by 4, 16 is divisible by 4, so 32 + 16 is divisible by 4.
Divisibility of the difference. If the deducted and deducted are divisible by some number, then the difference is also divisible by this number.
Example 1.2
777 is divisible by 7, 49 is divisible by 7, so the difference 777 - 49 is divisible by 7.
Divisibility of a product by a number. If in the product at least one of the factors is divisible by some number, then the product is also divisible by this number.
Example 1.3
15 is divisible by 3, so the product 15 ∙ 17 ∙ 23 is divided 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 divided by 30, 30 = 2 ∙ 3 ∙ 5, so 30 is divided by 2, and 3, and 5.
B. Pascal made a great contribution to the study of signs of divisibility of numbers.Blaise Pascal (Blaise Pascal) (1623-1662), French religious thinker, mathematician and physicist, one of the greatest minds of the 17th century.He formulated the following divisibility criterion, which bears his name:
Natural number but divides by another natural number b only if the sum of the products of the digits of the number but by the corresponding residuals obtained by dividing bit units by the number b , is divisible by this number.
1.1 Divisibility criteria by 2, 5, 10, 3 and 9
At school, we study signs of divisibility by 2, 3, 5, 9, 10.
Divisibility by 10. All those and only those numbers, the record of which ends with the digit 0, are divided by 10.
Divisibility by 5. All those and only those numbers, the record of which ends with the digit 0 or 5, are divided by 5.
Divisibility by 2. All those and only those numbers whose recording ends with an even digit are divided by 2: 2,4,6,8 or 0.
Divisibility by 3 and 9. All those and only those numbers, the sum of the digits of which is divisible by 3 or 9, respectively, are divided by 3 and 9.
By writing the number (by its last digits), you can also set the divisibility of the number by 4, 25, 50, 8 and 125.
1.2 Divisibility criteria by 4, 25 and 50
Those and only those numbers that end in two zeros or in which the last two digits express a number divisible by 4, 25 or 50, respectively, are divisible by 4, 25 or 50.
Example 1.2.1
The number 97300 ends with two zeros, which means that it is divisible by 4, 25, and 50.
Example 1.2.2
The number 81764 is divisible by 4, since 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 by 50, since the number formed by the last two digits 50 is divisible by both 25 and 50.
1.3 Divisibility criteria by 8 and 125
Those and only those numbers that end in three zeros or in which the last three digits express a number divisible by 8 or 125, respectively, are divisible by 8 or 125.
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, since the number formed by the last three digits 864 is divisible by 8.
Example 1.3.3
The number 179 250 is divisible by 125, since the number formed by the last three digits 250 is divisible by 125.
1.4 Simplification of the divisibility criterion by 8
The question of divisibility of some number is reduced to the question of divisibility by 8 of some 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 have to actually make a division.
The question naturally arises: is it possible to simplify the criterion for divisibility by 8 as well? You can, 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, in which the two-digit number formed by the digits of hundreds and tens, added with half the number of ones, is divisible by 4.
Example 1.4.1
Is 592 divisible by 8?
Solution.
We separate units from the number 592 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 that the number 592 is divisible by 8.
Answer: it is divided.
1.5 Divisibility criteria by 6, 12, 15, 18, 45, etc.
Using the property of divisibility of a number by a product, from the above criteria for divisibility, we obtain criteria for divisibility by 6, 12, 15, 18, 24, etc.
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 31,242 is divisible by 6, as it is divisible by both 2 and 3.
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 316,224 is divisible by 12 because it is divisible by both 4 and 3.
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 812 445 is divisible by 15, as it is divisible by both 3 and 5.
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 817,254 is divisible by 18, as it is divisible by both 2 and 9.
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, as it is divisible by both 5 and 9.
There is another sign that numbers are divisible by 6.
1.6 Divisibility by 6
To check the divisibility of a number by 6, you need:
Hundreds multiplied by 2
Subtract the result from the number after the number of hundreds.
If the result is divisible by 6, then the whole number is divisible by 6. Example 1.6.1
Is 138 divisible by 6?
Solution.
The number of hundreds is 1 2 = 2, 38-2 = 36, 36: 6, which means 138 is divided 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 Divisibility by 7
To find out if a number is divisible by 7, you need:
The number worth up to tens multiplied by two,
Add the remaining number to the result.
Check if the result is divisible by 7 or not.
Example 2.1.1
Is 4 690 divisible by 7?
Solution.
A number worth up to tens of 46 2 = 92, 92 + 90 = 182, 182: 7 = 26, so 4690 is divisible by 7.
2.2 Divisibility by 11
The number is divisible by 11 if the difference between the sum of the digits in the odd places and the sum of the digits in the even places is a multiple of 11.
The difference can be negative or zero, but must be a multiple of 11.
Example 2.2.1
Is 100397 divisible by 11?
Solution.
The sum of the numbers in even places: 1 + 0 + 9 = 10.
The sum of the digits in the odd places: 0 + 3 + 7 = 10.
The difference between the sums: 10 - 10 = 0, 0 is a multiple of 11, which means that 100397 is divisible by 11.
2.3 Divisibility by 13
A number is divisible by 13 if and only if the result of subtracting the last digit multiplied by 9 from this 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 Divisibility criteria by 19
A number is divisible by 19 without a remainder when the number of its tens, added with twice the number of ones, is divisible by 19.
Example 2.4.1
Determine if 1026 is divisible by 19.
Solution.
In the number 1026 there are 102 tens and 6 units. 102 + 2 ∙ 6 = 114;
Similarly 11 + 2 ∙ 4 = 19.
As a result of performing two consecutive steps, we got the number 19, which is divisible by 19, therefore, the number 1026 is divisible by 19.
The combined criterion for divisibility by 7, 11 and 13
In the table of prime numbers, the numbers 7, 11 and 13 are located side by side. Their product is: 7 ∙ 11 ∙ 13 = 1001 = 1000 + 1. Hence, 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 numbers as the multiplicable, 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 of the question of divisibility of a multi-digit number by 7 or 11, or 13 to the divisibility by them of some other number - no more than three-digit.
If the difference between the sums of the faces of a given number, taken through one, is divisible by 7 or 11, or 13, then this number is divisible, respectively, by 7 or 11, or 13.
Example 3.1
Determine if 42 623 295 is divisible by 7, 11 and 13.
Solution.
We split this number from right to left into 3-digit edges. The leftmost edge may or may not have three digits. 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, 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 was very fond of the people and was included in his songs and sayings:
Try on seven times, cut 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 criteria for divisibility by 7.
Let us explain the first criterion of divisibility by 7 using 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 this number is also divisible (not divisible) by 7.
Since 168 is divisible by 7, 5236 is also 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 the next digit; multiply the result by 3 and add the next digit, and so on up to the last digit. For simplicity, 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 also 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 the multiplication should not start from the leftmost digit of the given number, but from the extreme right 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; the addition of the number 7 can be skipped, since the number 7 is subtracted from the result; 5 ∙ 5 = 25; (25 - 21 = 4); 4 + 3 = 7 - is divisible by 7, which means that 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, and so on, alternating between 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 test number is also 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 - is divisible by 7, which means that the number 889 is divisible by 7.
And more signs of divisibility by 7. If any two-digit number is divisible by 7, then it is divided by 7 and the number reversed, increased by the tens digit of the given number.
Example 4.3
14 is divisible by 7, so it is divisible by 7 and 41 + 1.
35 is divisible by 7, so 53 + 3 is divisible by 7.
If any three-digit number is divisible by 7, then it is divided by 7 and the number reversed, reduced by the difference between the digits of ones and hundreds of the given number.
Example 4.4
The number 126 is divisible by 7. Therefore, the number 621 - (6 - 1), that is, 616, is divisible by 7.
Example 4.5
The number 693 is divisible by 7. Therefore, it is divisible by 7 and the number 396 is (3 - 6), that is, 399.
Extending Divisibility 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 the divisibility of a given number by 13, 17 or 19, it is necessary to multiply the leftmost digit of the tested number by 3, 7 or 9, respectively, and subtract the next digit; the result is again multiplied by 3, 7 or 9, respectively, and add the next digit, etc., alternating subtraction and addition of 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 thereof.
If the final result is divisible (not divisible) by 13, 17 and 19, then this number is also divisible (not divisible).
Example 5.1
Is 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 that 2,075,427 is divisible by 19.
Generalized divisibility criterion
The idea of cutting a number on a face and then adding them up to determine the divisibility of a given number turned out to be very fruitful and led to a uniform criterion for the divisibility of multi-digit numbers into a rather extensive group of primes. One of the groups of "lucky" divisors are all integer factors p of the number d = 10n + 1, where n = 1, 2, 3,4, ... (for large values of n, the practical meaning of the feature is lost).
101
101
1001
7, 11, 13
10001
73, 137
2) fold the edges through one, starting from the extreme right;
3) fold the rest of the edges;
4) deduct the smaller one from the larger amount.
If the result is divisible by p, then this number is also divisible by p.
So, to determine the divisibility of a number by 11 (p = 11), we cut the number on the face by one digit (n = 1). Proceeding further, as indicated, we arrive at the well-known criterion for divisibility by 11.
When determining the divisibility of a number by 7, 11 or 13 (p = 7, 11, 13), cut off 3 digits (n = 3). When determining the divisibility of a number by 73 and 137, we cut off 4 digits (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 on the edge: 837 3621 7250 4831.
We fold the edges through one: 4931 + 3621 = 8452; 7250 + 837 = 8087.
Subtract the smaller one from the larger amount: 8452-8087 = 365.
365 is divisible by 73, but not divisible by 137; hence, this 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 factors:
n
d
p
1
9
3
3
999
37
5
99 999
41, 271
To determine the divisibility of any number by any of these numbers p you need:
1) cut the given number from right to left (from ones) on the face of n digits (each p corresponds to its own n; the extreme left face can have digits less than n);
2) fold all the edges.
If the result is divisible (not divisible) by p, then the given number is also 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 divisible by 37.
Likewise: 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, myknowledge of mathematics. II learned that apart from the signs I know for 2, 3, 5, 9 and 10, 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 a place for creativity.
The work is theoretical andpractical use... This research will be useful in preparation 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 this or that sign to a specific task, apply the studied signs in a real situation. In the future, I propose to continue working on the study of divisibility criteria for numbers.
Literature
1. NN Vorobyov "Signs of divisibility" Moscow "Science" 1988
2. K. I. Shchevtsov, G. P. Bevz "Handbook of elementary mathematics" Kiev "Naukova Dumka" 1965
3. M. Ya. Vygodsky "Handbook of elementary mathematics" Moscow "Science" 1986
4. Internet resources
To simplify the division of natural numbers, the rules for division into the numbers of the first ten and the numbers 11, 25 were derived, which are combined into a section divisibility criteria for 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 2, 3, 4, 5, 6, 9, 10, 11, 25 and a digit unit?
Natural numbers that have digits in the first digit (ending in) 2,4,6,8,0 are called even.
Divisibility of numbers by 2
All even natural numbers are divisible by 2, for example: 172, 94.67 838, 1670.
Divisibility of numbers by 3
All natural numbers are divided by 3, the sum of the digits of which is a multiple of 3. For example:
39 (3 + 9 = 12; 12: 3 = 4);
16 734 (1 + 6 + 7 + 3 + 4 = 21; 21:3 = 7).
Divisibility of numbers by 4
All natural numbers are divided 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 of numbers by 5
Divisibility of numbers by 6
Those natural numbers that are divisible by 2 and 3 at the same time are divided by 6 (all even numbers that are divisible by 3). For example: 126 (b - even, 1 + 2 + 6 = 9, 9: 3 = 3).
Divisibility of numbers by 9
Those natural numbers whose sum of digits is divisible by 9 is divided by 9. For example:
1179 (1 + 1 + 7 + 9 = 18, 18: 9 = 2).
Divisibility of numbers by 10
Divisibility of numbers by 11
Only those natural numbers are divided 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 of numbers by 25
Those natural numbers whose last two digits are zeros or are divisible by 25 are divisible by 25. For example:
2 300; 650 (50: 25 = 2);
1 475 (75: 25 = 3).
Divisibility of numbers per bit unit
Those natural numbers are divided by the bit unit, for which the number of zeros is greater than or equal to the number of zeros of the bit unit. For example: 12,000 is divisible by 10, 100 and 1000.
Divisibility tests for numbers- these are the rules that allow, without making division, relatively quickly find out whether this number is divisible by a given one without a remainder.
Some of divisibility criteria quite simple, some more difficult. On this page you will find both the divisibility criteria for prime numbers, such as, for example, 2, 3, 5, 7, 11, and the divisibility criteria for composite numbers, such as 6 or 12.
I hope this information will be useful to you.
Happy learning!
Divisibility by 2
This is one of the simplest divisibility criteria. It sounds like this: if the recording of a natural number ends in an even digit, then it is even (divisible by 2 without a remainder), and if the recording of a number ends in an odd digit, then this number is odd.
In other words, if the last digit of the 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.
And numbers: 23 5
, 137
, 2303
are not divisible by 2 because they are odd.
Divisibility by 3
This divisibility criterion has completely different rules: if the sum of the digits of a number is divisible by 3, then the number is also 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 either.
So, in order to understand whether a number is divisible by 3, you just need to add together the numbers of which it consists.
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 without ostak), and in the second 1 + 4 + 1 = 6
(6: 3 = 2 - also divisible by 3 without ostak).
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).
Divisibility by 4
This divisibility criterion will be more complicated. If the last 2 digits of the number form a number that is divisible by 4 or it is 00, then the number is divisible by 4, otherwise this number is not divisible by 4 without a remainder.
For example: 1 00
and 3 64
are divided 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
nor 86
are not divisible by 4, which means they do not correspond to the given criterion of divisibility.
Divisibility by 5
And again we have a rather simple divisibility sign: if the record of a natural number ends with a digit 0 or 5, then this number is divisible without a remainder by 5. If the record 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
e.g. 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 in 5 or 0, which means they cannot be divisible by 5 without a remainder.
Divisibility by 6
Before us is a composite number 6, which is the product of the numbers 2 and 3. Therefore, the divisibility by 6 is also composite: in order for a number to be divisible by 6, it must correspond to two divisibility features at the same time: the divisibility feature by 2 and the divisibility feature by 3. At the same time, 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 back to the divisibility by 6 criterion.
The numbers 138 and 474 are even and correspond to the criteria of 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 divisibility criterion by 2, and therefore do not correspond to the divisibility criterion by 6.
Divisibility by 7
This divisibility criterion is more complex: a number is divisible by 7 if the result of subtracting the last doubled digit from the tens of this number is divisible by 7 or equal to 0.
Sounds pretty confusing, but simple in practice. See for yourself: the number 95
9 is divisible by 7 because 95
-2 * 9 = 95-18 = 77, 77: 7 = 11 (77 is divisible by 7 without remainder). Moreover, if difficulties arose with the number obtained during the transformations (because of 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 signs 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's hard for us to understand if 457
8 by 7, so let's repeat the process: 457
-2 * 8 = 457-16 = 441. And again we will use the divisibility criterion, since we still have a three-digit number 44
1. So, 44
-2 * 1 = 44-2 = 42, 42: 7 = 6, i.e. 42 is divisible by 7 without remainder, which means 45801 is divisible by 7.
But the numbers 11
1 and 34
5 is not divisible by 7 because 11
-2 * 1 = 11 - 2 = 9 (9 is not evenly divisible by 7) and 34
-2 * 5 = 34-10 = 24 (24 is not evenly divisible by 7).
Divisibility by 8
Divisibility by 8 is as follows: if the last 3 digits form a number that is divisible by 8, or 000, then the given number is divisible by 8.
Numbers 1 000
or 1 088
divisible by 8: the first ends in 000
, the second 88
: 8 = 11 (divisible by 8 without remainder).
But the numbers 1 100
or 4 757
are not divisible by 8, since the numbers 100
and 757
are not evenly divisible by 8.
Divisibility 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 also 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 either.
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 ostak), and in the second 1 + 4 + 4 = 9
(9: 9 = 1 - also divisible by 9 without an ostak).
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 is divisible by 9 without a remainder).
Divisibility by 10, 100, 1000 and other bit units
I combined these divisibility signs because they can be described in the same way: a number is divided by a bit unit if the number of zeros at the end of the number is greater than or equal to the number of zeros in a given bit unit.
In other words, for example, we have numbers like this: 654 0
, 46400
, 867000
, 6450
... of which all are divisible by 1 0
; 46400
and 867 000
are also divided by 1 00
; and only one of them - 867 000
divisible by 1 000
.
Any numbers that have less zeros at the end than a bit unit are not divisible by that bit unit, for example 600 30
and 7 93
not divisible 1 00
.
Divisibility by 11
In order to find out whether a number is divisible by 11, you need to get 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 propose to consider 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.
But 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 by 12
The number 12 is compound. Its divisibility criterion is the correspondence to the divisibility criteria 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 and the first and three times the number is divisible by 3), and znit, they are divisible by 12 without a remainder.
But 200 or 630 are not divisible by 12, because in the first case the number corresponds only to the sign of divisibility by 4, and in the second - only to the sign of divisibility by 3. but not to both signs at the same time.
Divisibility by 13
The sign of divisibility by 13 is that if the number of tens of a number, added with multiplied by 4 units of this number, is a multiple of 13 or equal to 0, then the number itself is divisible by 13.
Take for example 70
2. So, 70
+ 4 * 2 = 78, 78: 13 = 6 (78 is divisible by 13 without remainder), which means 70
2 is divisible by 13 without remainder. Another example is the number 114
4. 114
+ 4 * 4 = 130, 130: 13 = 10. The number 130 is divisible by 13 without a remainder, which means that the given number corresponds to the divisibility criterion 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, which means that the given numbers are not evenly divisible by 13.
Divisibility of numbers
As can be seen from the above, we can assume that for any of the natural numbers you can choose your own individual divisibility criterion or a "composite" feature if the number is a multiple of several different numbers. But as practice shows, in general, the larger the number, the more complex its sign. Perhaps the time spent on checking the divisibility criterion may turn out to be equal to or more than the division itself. Therefore, we usually use the simplest of the divisibility criteria.
