How many numbers are divisible by 13. Now I would like to show several other signs of divisibility and not only for prime numbers, but also for composite ones
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 and not only on prime numbers, but also into components.
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.
The number is divisible by 2 if and only if its last digit is divisible by 2, that is, it is even.
For example:
2, 8, 16, 24, 66, 150 - divisible by 2
, since the last digit of these numbers is even;
3, 7, 19, 35, 77, 453 - not divisible by 2
, since the last digit of these numbers is odd.
Test for divisibility by 3
The number is divisible by 3 if and only if the sum of its digits is divisible by 3.
For example:
471 - divisible by 3
, since 4+7+1=12, and the number 12 is divisible by 3;
532 - not divisible by 3
, since 5+3+2=10, and the number 10 is not divisible by 3.
Test for divisibility by 4
The number is divisible by 4 if and only if its last two digits form a number that is divisible by 4. A two-digit number is divisible by 4 if and only if twice the number of tens added to the number of units is divisible by 4.
For example:
4576 - divisible by 4
, since the number 76 (7·2+6=20) is divisible by 4;
9634 - not divisible by 4
, since the number 34 (3·2+4=10) is not divisible by 4.
Divisibility test by 5
The number is divisible by 5 when the last digit is divisible by 5, i.e. if it is 0 or 5.
For example:
375, 5680, 233575 - divided by 5
, since their last digit is 0 or 5;
9634, 452, 389753 - not divisible by 5
, since their last digit is not 0 or 5.
Divisibility test by 6
The number is divisible by 6 if and only if it is divisible by both 2 and 3, that is, if it is even and the sum of its digits is divisible by 3.
For example:
462, 3456, 24642 - divisible by 6
, since they are divisible by both 2 and 3 at the same time;
6
, since 861 is not divisible by 2, 3458 is not divisible by 3, 34681 is not divisible by 2.
Test for divisibility by 7
The number is divisible by 7, if the difference between the tens digit and double the ones digit is divisible by 7.
For example:
Number 296492
We take the last digit “2”, double it, we get 4. Subtract 29649-4=29645. We don't know if it is divisible by 7. So let's check again.
We take the last digit “5”, double it, we get 10. Subtract 2964-10=2954. We don't know if it's divisible by 7. So let's check again.
We take the last digit “4”, double it, we get 8. Subtract 295-8=287. We don't know if it is divisible by 7. So let's check again.
We take the last digit “7”, double it, we get 14. Subtract 28-14=14. The number 14 is divisible by 7, which means the original number is divisible by 7
Divisibility test by 8
The number is divisible by 8 if and only if the number formed by its last three digits is divisible by 8. Three digit number is divisible by 8 if and only if the number of units added to twice the number of tens and quadruple the number of hundreds is divisible by 8.
For example:
952 is divisible by 8 since 9*4+5*2+2=48 is divisible by 8
Divisibility test by 9
The number is divisible by 9 if and only if the sum of its digits is divisible by 9.
For example:
468, 4788, 69759 - divided by 9
, since the sum of their digits is divisible by nine (4+6+8=18, 4+7+8+8=27, 6+9+7+5+9=36);
861, 3458, 34681 - not divisible by 9
, since the sum of their digits is not divisible by nine (8+6+1=15, 3+4+5+8=20, 3+4+6+8+1=22).
Divisibility test by 10
The number is divisible by 10 if and only if it ends in zero.
For example:
460, 24000, 1245464570 - divided by 10
, since the last digit of these numbers is zero;
234, 25048, 1230000003 - not divisible by 10
, since the last digit of these numbers is not zero.
Divisibility test by 11
Sign 1: the number is divisible by 11 if and only if the modulus of the difference between the sum of the digits occupying odd positions and the sum of the digits occupying even positions is divisible by 11.
For example, 9163627 is divisible by 11 because it is divisible by 11.
Another example is that 99077 is divisible by 11 because it is divisible by 11.
Sign 2: the number is divisible by 11 if and only if the sum of numbers forming groups of two digits (starting with ones) is divisible by 11.
For example, 103785 is divisible by 11, since 11 is divisible by
Divisibility test by 13
Sign 1: The number is divisible by 13 when the sum of the number of tens and quadruple the number of ones is divisible by 13.
For example, 845 is divisible by 13, since 13 is divisible by
Sign 2: The number is divisible by 13 then, when the difference between the number of tens and nine times the number of units is divided by 13.
For example, 845 is divisible by 13, since 13 is divisible
Divisibility test by 17
The number is divisible by 17 when the modulus of the difference between the number of tens and five times the number of ones is divided by 17.
The number is divisible by 17 when the modulus of the sum of the number of tens and the number twelve multiplied by the number of units is divided by 17.
For example, 221 is divisible by 17 because it is divisible by 17.
Divisibility test by 19
The number is divisible by 19 if and only if the number of tens added to twice the number of units is divisible by 19.
For example, 646 is divisible by 19, since 19 is divisible by 19.
Divisibility test by 20
The number is divisible by 20 if and only if the number formed by the last two digits is divisible by 20.
Another wording: the number is divisible by 20 if and only if the last digit of the number is 0, and the second to last digit is even.
Tests for divisibility by 23
Sign 1: the number is divisible by 23 if and only if the number of hundreds added to triple the number formed by the last two digits is divisible by 23.
For example, 28842 is divisible by 23, since 23 is also divisible by
Sign 2: the number is divisible by 23 if and only if the number of tens added to seven times the number of ones is divisible by 23. For example, 391 is divisible by 23 because it is divisible by 23.
Sign 3: the number is divisible by 23 if and only if the number of hundreds added to seven times the number of tens and triple the number of ones is divisible by 23.
For example, 391 is divisible by 23 because it is divisible by 23.
Test for divisibility by 25
The number is divisible by 25 if and only if its last two digits form a number that is divisible by 25.
Test for divisibility by 27
The number is divisible by 27 if and only if the sum of numbers forming groups of three digits (starting with ones) is divisible by 27.
Test for divisibility by 29
The number is divisible by 29 if and only if the number of tens added to three times the number of units is divisible by 29.
For example, 261 is divisible by 29 because it is divisible by 29.
Divisibility test by 30
The number is divisible by 30 if and only if it ends in 0 and the sum of all digits is divisible by 3.
For example: 510 is divisible by 30, but 678 is not.
Test for divisibility by 31
The number is divisible by 31 if and only if the modulus of the difference between the number of tens and three times the number of ones is divisible by 31. For example, 217 is divisible by 31 because it is divisible by 31.
Test for divisibility by 37
Sign 1: the number is divisible by 37 if and only if, when dividing a number into groups of three digits (starting with ones), the sum of these groups is a multiple of 37.
Sign 2: the number is divisible by 37 if and only if the modulus of three times the number of hundreds added to four times the number of tens minus the number of units multiplied by seven is divisible by 37.
Sign 3: the number is divisible by 37 if and only if the modulus of the sum of the number of hundreds with the number of ones multiplied by ten minus the number of tens multiplied by 11 is divisible by 37.
For example, the number 481 is divisible by 37, since 37 is divisible by
Test for divisibility by 41
Sign 1: the number is divisible by 41 if and only if the modulus of the difference between the number of tens and four times the number of units is divisible by 41.
For example, 369 is divisible by 41 because it is divisible by 41.
Sign 2: to check whether a number is divisible by 41, it should be divided from right to left into edges of 5 digits each. Then in each face, multiply the first digit on the right by 1, multiply the second digit by 10, the third by 18, the fourth by 16, the fifth by 37, and add all the resulting products. If the result is divisible by 41, then and only then will the number itself be divisible by 41.
Divisibility test by 50
The number is divisible by 50 if and only if the number formed by its two lowest decimal digits is divisible by 50.
Test for divisibility by 59
The number is divisible by 59 if and only if the number of tens added to the number of ones multiplied by 6 is divisible by 59. For example, 767 is divisible by 59, since 59 is divisible by
Test for divisibility by 79
The number is divisible by 79 if and only if the number of tens added to the number of units multiplied by 8 is divisible by 79. For example, 711 is divisible by 79, since 79 is divisible by .
Divisibility test by 99
The number is divisible by 99 if and only if the sum of numbers forming groups of two digits (starting with ones) is divisible by 99. For example, 12573 is divisible by 99 because 99 is divisible by
Test for divisibility by 101
The number is divisible by 101 if and only if the modulus of the algebraic sum of numbers forming odd groups of two digits (starting with ones), taken with the “+” sign, and even numbers with the “-” sign, is divisible by 101.
For example, 590547 is divisible by 101 because 101 is divisible by
Divisibility test
Divisibility sign- a rule that allows you to relatively quickly determine whether a number is a multiple of a predetermined number without having to do the actual division. As a rule, it is based on actions with part of the digits from the number written in the positional number system (usually decimal).
There are several simple rules that allow you to find small divisors of a number in the decimal number system:
Test for divisibility by 2
Test for divisibility by 3
Test for divisibility by 4
Divisibility test by 5
Divisibility test by 6
Test for divisibility by 7
Divisibility test by 8
Divisibility test by 9
Divisibility test by 10
Divisibility test by 11
Divisibility test by 12
Divisibility test by 13
Divisibility test by 14
Divisibility test by 15
Divisibility test by 17
Divisibility test by 19
Test for divisibility by 23
Test for divisibility by 25
Divisibility test by 99
Let's divide the number into groups of 2 digits from right to left (the leftmost group can have one digit) and find the sum of these groups, considering them two-digit numbers. This sum is divisible by 99 if and only if the number itself is divisible by 99.
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).
Test for divisibility by 2 n
The number is divisible by nth power twos if and only if the number formed by its last n digits is divisible by the same power.
Divisibility test by 5 n
A number is divisible by the nth power of five if and only if the number formed by its last n digits is divisible by the same power.
Divisibility test by 10 n − 1
Let's divide the number into groups of n digits from right to left (the leftmost group can have from 1 to n digits) and find the sum of these groups, considering them n-digit numbers. This amount is divided by 10 n− 1 if and only if the number itself is divisible by 10 n − 1 .
Divisibility test by 10 n
A number is divisible by the nth power of ten if and only if its last n digits are
Zoloeva Alana, Volkova Nicole
The work shows methods of graphic multiplication of multi-digit numbers, as well as signs of divisibility by 7, 11, 13, 17,19.23, etc. Can be used in mathematics lessons in grades 5-6.
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Slide captions:
Mathematics project “In the world of numbers” On the topic: “Signs of divisibility” by 6B grade students of Lyceum No. 597 Alana Zoloeva and Nicole Volkova
About the signs of divisibility The test of divisibility is a rule that allows you to quickly determine the multiple of a given number. Since antiquity and simple people, and scientists were interested in signs of divisibility of numbers and found them. But a special contribution to the study of divisibility tests was made by the French mathematician Blaise Pascal.
Pascal's test Pascal's test is a test of divisibility for all natural numbers, that is, division. Also, Blaise Pascal discovered signs of divisibility of natural numbers by certain integers. Any number a will be divided by any number b only if the sum of the products of the digits of the number a by the corresponding remainders obtained by dividing the digit units by the number b is divisible by this number.
Graphic multiplication. Double figures. There is a very convenient way to multiply, graphic multiplication. Let's say we need to multiply 32 by 21. We draw lines, starting with the number 32. We draw 3 lines diagonally from the upper right corner to the lower left, and a little lower, parallel to them, 2 lines. Then the number 21: draw 2 lines on the left, lower, and 1 on the right, higher. We mark the points of intersection of the lines, count them in each “zone”, and get the result. We get this diagram:
Graphic multiplication. Multi-digit numbers. Multi-digit numbers are multiplied graphically in the same way as two-digit numbers, but the sum of the points in 1 “zone” is often two-digit numbers. In such cases, the first digit double digit number is added to the previous number. For example, we multiply 123 by 412. Here's the diagram:
Number 4 Natural three-digit and large numbers are divisible by 4 only when their last two digits are zeros or multiples of 4. For example, the number 497764. It is divisible by 4, since 64 is divisible by 4, that is, the last 2 digits of this number. Two-digit natural numbers are divisible by 4 only when the sum of twice the tens number and the ones number is divisible by 4. Take the same number 64. 6∙2= 12 and + 4 = 16, so 64 is divisible by 4.
Number 6 A number is divisible by 6 when it is divisible by both 2 and 3 at the same time, and also when the quadruple number of tens, when added to the number of units, is divisible by 6. For example, the number 144 is divisible by 6, since 14∙4+4=60 is divisible by 6.
Number 7 A number is divisible by 7 when triple the number of tens added to the number of ones is divisible by 7. For example, the number 154 is divisible by 7, since 15∙3+4=49 is divisible by 7.
Number 8 A three-digit number is divisible by 8 if and only if the number of units added to twice the number of tens and quadruple the number of hundreds is divisible by 8. For example, the number 952 is divisible by 8, since 2+5∙2+9∙4=48 is divisible by 8.
Number 11 1 sign: a number is divisible by 11 when the modulus of the difference between the sum of the digits occupying an odd position and occupying an even one is divisible by 11. For example, the number is 10538. 1+5+8=14, 0+3=3, 14-3=11. │11 │ =11, and 11 is divisible by 11, which means the number 10538 is also divisible by 11. 2nd sign: a number is divisible by 11 when the sum of numbers forming groups of 2 digits, starting with ones, is divisible by 11. For example, the number 10593. 93+5+1=99. 99 is a multiple of 11, which means the number 10593 is also a multiple of 11.
Number 13 A number is divisible by 13 when the sum of the number of tens and quadruple the number of ones is divisible by 13. For example, the number 845 is divisible by 13, since 84+5∙4=104 and 10+4∙4=26 are divisible by 13.
Number 17 A number is divisible by 17 when the modulus of the difference between the number of tens and five times the number of ones is divisible by 17. For example, the number 221 is divisible by 17 because | 22-5 ∙ 1 |=17 .
Number 19 A number is divisible by 19 when the number of tens added to twice the number of ones is divisible by 19. For example, the number 646 is divisible by 19, since 64+6*2=76 and 7+6*2=19.
Number 23 A number is divisible by 23 when the number of tens added to seven times the number of ones is divisible by 23. For example, the number 391 is divisible by 23, since 39+1 ∙ 7=46 is divisible by 23.
Number 25 A number is divisible by 25 when the number formed by its last 2 digits is divisible by 25. For example, the number 1765375. 75 is divisible by 25, which means given number also a multiple of 25.
Number 99 A number is divisible by 99 when the sum of numbers forming groups of 2 digits, starting with ones, is divisible by 99. For example, the number 64449. 49+44+6=99. 99 is a multiple of 99, therefore 64449 is a multiple of 99.
Number 101 A number is divisible by 101 when the modulus of the sum of numbers forming odd groups of 2 digits, (starting from ones) taken with the “+” sign and forming even groups of 2 digits, taken with the “-” sign, is divisible by 101. For example, the number is 363297. │97+36-32 │=101, which means this number is a multiple of 101.
Thank you for your attention!
