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 characteristics 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 adding 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 whose result is the addition of digits located in odd places, or equal to the sum numbers placed in even places, or different by 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.

In the table of prime numbers, that is, those that are divisible only by 1 and themselves, the numbers 7, 11 and 13 are located next to each other (see the table of prime numbers on page 363). Their product is equal to:

7 ∙ 11 ∙ 13=1001 = 1000 + 1.

Let us note for now that 1000 + 1 is divisible by 7, 11, and 13. Further, if any three digit number multiplied by 1001, then the product will be written in the same numbers as the multiplicand, only repeated twice.

Let
- any three-digit number (a, b and c are the digits of this number). Let's multiply it by 1001:

Consequently, all numbers of the form abcabc are divisible by 7, 11 and 13. In particular, the number 999,999, or, otherwise, 1000,000-1, is divisible by 7, 11 and 13.

The indicated patterns allow us to reduce the solution to the question of the divisibility of a multi-digit number by 7 or 11,

or by 13 to the divisibility of some other number by them - no more than three-digit.

Let's say we need to determine whether the number 42,623,295 is divisible by 7, 11 and 13. Let's divide this number from right to left into faces of 3 digits. The leftmost edge may not have three digits. Let us now present this number in this form:

42 623 295 = 295 + 628 ∙ 1000 + 42 ∙ 1 000 000,

or (similar to what we did when considering the test for divisibility by 11):

42 623 295 = 295 + 623 (1000 + 1 -1) + 42(1000000 - 1 + 1) = (295 - 623 + 42) + .

The number in square brackets is necessarily divisible by 7, 11, and 13. This means that the divisibility of the tested number by

7, 11 and 13 are entirely determined by the divisibility of the number enclosed in the first parenthesis.

Considering each face of the tested number as an independent number, we can state the following combined sign of divisibility into three numbers at once, 7, 11 and 13:

Calculate the difference between the sums of the faces given number, taken through one, is divisible by 7 or 11, or 13, then this number is divisible by 7 or 11 or 13, respectively.

Let's return to the number 42,623,295. Let's determine which of the numbers 7, 11 or 13 divides the difference between the sums of the faces of this number:

(295 + 42)-623 = -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.

Obviously, the divisibility by 7, 11 and 13 of four-, five- and six-digit numbers, that is, numbers that are divided into only 2 sides (practically the more common case), is determined by the divisibility by 7, 11 and 13 of the difference between the sides of a given number. So, for example, it is easy to establish that 29,575 is divisible by 7 and 13, but not divisible by 11. Indeed, the difference between the faces is

and the number 546 is divisible by 7 and 13 and not divisible by 11.

Task. By establishing the combined divisibility criterion for 7, 11 and 13, we operated with a number that was divided into 3 faces. Justify this feature using the example of a number that is divided into 4 faces of 3 digits from right to left.

Commenting is closed now!

While working at a preparatory school, I walked into the sixth-grade office and saw a poster on the wall “Signs of divisibility of numbers.” There were signs of divisibility for the numbers 2, 3, 4, 5, 6, 8, 9, but for the number 7 there was no such sign. I asked the math teacher:

— Why is there no sign of divisibility by seven?

They told me that it exists, but it is very complicated. I made inquiries on the Internet. I found three signs.

Sign 1 : the number is divisible by if and only if triple the number of tens added to the number of ones is divisible by 7. For example, 154 is divisible by 7, since 15*3+4=49 is divisible by 7.

Another example is that the number 1001 is divisible by 7, since 100*3+1=301, 30*3+1=91, 9*3+1=28, 2*3+8=14 are divisible by 7.

Sign 2 . a number is divisible by 7 if and only if the modulus of the algebraic sum of numbers forming odd groups of three digits (starting with ones), taken with a “+” sign, and even numbers with a “-” sign is divisible by 7. For example, 138689257 is divisible by 7, since 7 is divisible by |138-689+257|=294.

Sign 3 . 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).

Let's check the divisibility of a number 86 576 (eighty six thousand five hundred seventy six). In this number 8 657 (eight thousand six hundred fifty seven) tens and 6 (six) units. Let's start checking the divisibility of this number by 7 (seven):

8657 - 6 x 2 = 8657 - 12 = 8645

Again we check divisibility by 7 (seven), now the number we have already received 8 645 (eight thousand six hundred forty-five). Now we have 864 (eight sixty four) tens and 5 (five) units:

864 - 5 x 2 = 864 - 10 = 854

We repeat our actions again for the number 854 (eight hundred fifty four), in which 85 (eighty five) tens and 4 (four) units:

85 - 4 x 2 = 85 - 8 = 77

In principle, it is already visible to the naked eye that the number 77 (seventy seven) divided by 7 (seven) and the result is 11 (eleven). We have already considered a similar result above.

As you can see, the signs are really complex. It is difficult to use them in your mind because large quantity operations. The simplest is the third sign, but there are also two actions, first multiplication and then subtraction, and for numbers over 700 you already need to do several cycles.

Set the task:

“Find division by 7 with fewer math operations.”

I used the TRIZ tool – IFR (ideal final result).

The number itself must provide a resource for calculation.

And this resource was found. If you look at the multiplication table for 7, then its products have a distinctive property - the final digit is not repeated: 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70. At first glance, this complicates the task, since .To. the number being checked with any ending can be divisible by 7. But according to the TRIZ rule: “Whoever interferes helps.” We must use this property to our advantage.

Looking at the last digit in the number being tested, we already know one sign of the answer - this is the number from the multiplication table that gives this tip. For example, if the number being tested is 154, then if it is divisible by 7, the last digit in the answer should be 2 (7x2=14), and if the number is 259, then the last digit of the answer should be 7 (7x7=49).

Here is the resource you need - this is the multiplication table by 7 - 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70.

We assume that we have it in memory. Now we use the action from the third (simplest) attribute - subtraction. We get new sign divisible by 7.

A number is divisible by 7 when the result of subtracting the first digit famous work of this number without the last digit is divisible by 7.

And now in simple words.

— We look at the number being checked, for example, the already known 259.

— It ends in 9. We take the resource from the multiplication table 49 . Its first digit is 4.

— Let’s subtract this number from 25. 25 – 4 = 21

— The answer is 21. So the number is divisible by 7. This is: 259: 7 = 37. The last digit is 7, as we expected.

A few more examples. Is 756 divisible by 7?

It ends in 6. Resource is 56. Subtract 75 - 5 = 70. The number is divided by 756: 7 = 108

Number 392. Ends in 2. Resource – 42. Subtract 39 -4 = 35. Divide 392: 7 = 56.

Number 571. Ends with 1. Resource – 21. Subtract 57 – 2 = 55. Not divisible.

Number 574. Ends in 4. Resource – 14. Subtract 57 – 1 = 56. Divide 574: 7 = 82

In this feature, we excluded one mathematical operation - multiplication.

Addition.

For numbers being tested greater than 700, to avoid repeated cycles, as in sign 3, use multiples of sevens for the subtrahend.

Consider, for example, the number 973. It ends in 3. Resource is 63. Subtract 97 - 6 = 91. You can go to the second cycle, or you can subtract not 6, but 76. 97 - 76 = 21. Divides.

Additions are made according to the number system of seven: 70, 140, 210, etc. depending on the number being checked.

1. This sign can be used mentally without much difficulty for numbers up to 1000. It will help you find multiples for division.

2. Colleagues, use TRIZ to solve your problems! This saves time. It took me 3 hours to find this sign of divisibility, taking into account the search for analogues on the Internet.

I will be glad if this sign is useful to someone.

Good afternoon
Today we will continue to look at signs of divisibility.
And we'll start with this:
We take the last digit of the number, double it and subtract it from the number that is left without this last digit. If the difference is divisible by 7, then the whole number is divisible by 7. This action can be continued as many times as desired until it becomes clear whether the number is divisible by 7 or not.

Example: 298109.
1st step. We take 9, multiply it by 2 and subtract:
29810-18=29792.
2nd step. 29792. Take 2, multiply it by 2 and subtract:
2979-4 = 2975.
3rd step. 2975. Take 5, multiply by 2 and subtract: 297-10=287.
4th step. 287. Take 7, multiply by 2 and subtract 28-14=14. Divisible by 7.
So the whole number 298109 is divisible by 7.

Another example. The number is 1102283.
1st step. 110228-3*2 = 110222
2nd step. 11022-2*2 = 11018.
3rd step. 1101-8*2 = 1085.
4th step. 108-5*2 = 98.
5th step. 9-8*2 = -7. Divisible by 7. So 1102283 is divisible by 7.

Test for divisibility by 13. We take the last digit of the number, multiply it by 4 and add it with the number without the last digit. If the sum is divisible by 13, then the whole number is divisible by 13.
This action can be continued as many times as desired until it becomes clear whether the number is divisible by 13 or not.
Example: Number 595166.
1st step. 59516 + 6*4 = 59540
2nd step. 5954 + 0*4 = 5954
3rd step. 595 + 4*4 = 611
4th step. 61 + 1*4 = 65
5th step. 6 + 5*4 = 26. Divisible by 13.
This means that the number 595166 is divisible by 13.

Another example. The number is 10221224.
1st step. 1022122 + 4*4 = 1022138
2nd step. 102213 + 8*4 = 102245
3rd step. 10224 + 5*4 = 10244
4th step. 1024 + 4*4 = 1040
5th step. 104 + 0*4 = 104
6th step. 10 + 4*4 = 26. Divisible by 13.
This means that the number 10221224 is divisible by 13.

Now I would like to show several other signs of divisibility, not only for prime numbers, but also for composite ones.

Test for divisibility by 11. Let's take a number and add up all the numbers that are in odd places. Then we add up all the digits of the number that are in even places.
If the difference between the first sum and the second is a multiple of 11, then the entire number is divisible by 11.
In this case, the difference can be either positive or negative.
Examples: 160369(Sum of digits that are in odd places
1+0+6 = 7. The sum of the numbers that are in even places is 6+3+9 = 18.
18 - 7 = 11. Divisible by 11. So the number 160369 is divisible by 11).
Another example: 7527927 (7+2+9+7 = 25. 5+7+2 = 14. 25 — 14 = 11.
The number 7527927 is divisible by 11).

Test for divisibility by 15. The number 15 is a composite number. It can be represented as a product of prime factors, namely 5 and 3.
And we already know. So, a number is divisible by 15 if
1. - it ends in 0 or 5;

Example: 36840(The number ends in 0; the sum of its digits is 3+6+8+4 = 21. Divisible by 3.) This means the whole number is divisible by 15.
Another example: 113445 The number ends in 5; the sum of its digits is 1+1+3+4+4+5 = 18. Divisible by 3.) This means the entire number is divisible by 15.

Test for divisibility by 12. The number 12 is composite. It can be represented as the product of the following factors: 4 and 3.
So a number is divisible by 12 if
1. - its last 2 digits are divisible by 4;
2. - the sum of its digits is divisible by 3.
Examples: 78864(The last two digits are 64. The number made up of them is divisible by 4; the sum of the digits is 7+8+8+6+4 = 33. Divisible by 3.) This means that the entire number is divisible by 12.
Another example: 943908(The last two digits are 08. The number made up of these digits is divisible by 4; the sum of the digits is 9+4+3+9+0+8 = 33.
Divisible by 3.) So the whole number is divisible by 12.

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).

Test for divisibility 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 specific value 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 by counting them double 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).