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 mentally due to the large number of 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 obtain a new test for divisibility 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.
Mathematics in 6th grade begins with studying the concept of divisibility and signs of divisibility. They are often limited to the criteria of divisibility by the following numbers:
- On 2 : last digit must be 0, 2, 4, 6 or 8;
- On 3 : the sum of the digits of the number must be divisible by 3;
- On 4 : the number formed by the last two digits must be divisible by 4;
- On 5 : last digit must be 0 or 5;
- On 6 : the number must have signs of divisibility by 2 and 3;
- Divisibility test for 7 often missed;
- They also rarely talk about the test of divisibility by 8 , although it is similar to the criteria for divisibility by 2 and 4. For a number to be divisible by 8, it is necessary and sufficient that the three-digit ending is divisible by 8.
- Divisibility test for 9 Everyone knows: the sum of the digits of a number must be divisible by 9. Which, however, does not develop immunity against all sorts of tricks with dates that numerologists use.
- Divisibility test for 10 , probably the simplest: the number must end in zero.
- Sometimes sixth graders are taught about the test of divisibility by 11 . You need to add the digits of the number that are in even places, and subtract the numbers that are in odd places from the result. If the result is divisible by 11, then the number itself is divisible by 11.
Let's take a number. We divide it into blocks of 3 digits each (the leftmost block can contain one or 2 digits) and alternately add/subtract these blocks.
If the result is divisible by 7, 13 (or 11), then the number itself is divisible by 7, 13 (or 11).
This method, like a number of mathematical tricks, is based on the fact that 7x11x13 = 1001. However, what to do with three-digit numbers, for which the question of divisibility also cannot be solved without division itself.
Using the universal test of divisibility, it is possible to construct relatively simple algorithms for determining whether a number is divisible by 7 and other “inconvenient” numbers.
Improved test for divisibility by 7
To check whether a number is divisible by 7, you need to discard the last digit from the number and subtract this digit twice from the resulting result. If the result is divisible by 7, then the number itself is divisible by 7.
Example 1:
Is 238 divisible by 7?
23-8-8 = 7. So the number 238 is divisible by 7.
Indeed, 238 = 34x7
This action can be carried out repeatedly.
Example 2:
Is 65835 divisible by 7?
6583-5-5 = 6573
657-3-3 = 651
65-1-1 = 63
63 is divisible by 7 (if we hadn’t noticed this, we could have taken one more step: 6-3-3 = 0, and 0 is certainly divisible by 7).
This means that the number 65835 is divisible by 7.
Based on the universal criterion of divisibility, it is possible to improve the criteria of divisibility by 4 and by 8.
Improved test for divisibility by 4
If half the number of units plus the number of tens is an even number, then the number is divisible by 4.
Example 3
Is the number 52 divisible by 4?
5+2/2 = 6, the number is even, which means the number is divisible by 4.
Example 4
Is the number 134 divisible by 4?
3+4/2 = 5, the number is odd, which means 134 is not divisible by 4.
Improved test for divisibility by 8
If you add twice the number of hundreds, the number of tens and half the number of units, and the result is divisible by 4, then the number itself is divisible by 8.
Example 5
Is the number 512 divisible by 8?
5*2+1+2/2 = 12, the number is divisible by 4, which means 512 is divisible by 8.
Example 6
Is the number 1984 divisible by 8?
9*2+8+4/2 = 28, the number is divisible by 4, which means 1984 is divisible by 8.
Divisibility test by 12- this is the union of the signs of divisibility by 3 and 4. The same works for any n that is the product of coprime p and q. For a number to be divisible by n (which is equal to the product pq,actih, such that gcd(p,q)=1), one must be divisible by both p and q.
However, be careful! For the compound divisibility criteria to work, the factors of a number must be coprime. You cannot say that a number is divisible by 8 if it is divisible by 2 and 4.
Improved test for divisibility by 13
To check whether a number is divisible by 13, you need to discard the last digit from the number and add it four times to the resulting result. If the result is divisible by 13, then the number itself is divisible by 13.
Example 7
Is 65835 divisible by 8?
6583+4*5 = 6603
660+4*3 = 672
67+4*2 = 79
7+4*9 = 43
The number 43 is not divisible by 13, which means that the number 65835 is not divisible by 13.
Example 8
Is 715 divisible by 13?
71+4*5 = 91
9+4*1 = 13
13 is divisible by 13, which means the number 715 is divisible by 13.
Signs of divisibility by 14, 15, 18, 20, 21, 24, 26, 28 and other composite numbers that are not powers of primes are similar to the tests for divisibility by 12. We check for divisibility by coprime factors of these numbers.
- For 14: for 2 and for 7;
- For 15: for 3 and for 5;
- For 18: on 2 and 9;
- For 21: on 3 and 7;
- For 20: by 4 and by 5 (or, in other words, the last digit must be zero, and the penultimate digit must be even);
- For 24: for 3 and for 8;
- For 26: on 2 and 13;
- For 28: on 4 and 7.
Instead of checking whether the 4-digit ending of a number is divisible by 16, you can add the ones digit with 10 times the tens digit, the quadruple hundreds digit, and the
multiplied by eight times the thousands digit and check if the result is divisible by 16.
Example 9
Is the number 1984 divisible by 16?
4+10*8+4*9+2*1 = 4+80+36+2 = 126
6+10*2+4*1=6+20+4=30
30 is not divisible by 16, which means 1984 is not divisible by 16.
Example 10
Is the number 1526 divisible by 16?
6+10*2+4*5+2*1 = 6+20+20+2 = 48
48 is not divisible by 16, which means 1526 is not divisible by 16.
An improved test for divisibility by 17.
To check whether a number is divisible by 17, you need to discard the last digit from the number and subtract this digit five times from the resulting result. If the result is divisible by 13, then the number itself is divisible by 13.
Example 11
Is the number 59772 divisible by 17?
5977-5*2 = 5967
596-5*7 = 561
56-5*1 = 51
5-5*5 = 0
0 is divisible by 17, which means the number 59772 is divisible by 17.
Example 12
Is the number 4913 divisible by 17?
491-5*3 = 476
47-5*6 = 17
17 is divisible by 17, which means the number 4913 is divisible by 17.
An improved test for divisibility by 19.
To check whether a number is divisible by 19, you need to add twice the last digit to the number remaining after discarding the last digit.
Example 13
Is the number 9044 divisible by 19?
904+4+4 = 912
91+2+2 = 95
9+5+5 = 19
19 is divisible by 19, which means the number 9044 is divisible by 19.
An improved test for divisibility by 23.
To check whether a number is divisible by 23, you need to add the last digit, increased by 7 times, to the number remaining after discarding the last digit.
Example 14
Is the number 208012 divisible by 23?
20801+7*2 = 20815
2081+7*5 = 2116
211+7*6 = 253
Actually, you can already notice that 253 is 23,
Rule
Test for divisibility by 7
To determine whether a number is divisible by \(\displaystyle 7\), you need to:
1. Take the original number without the last digit.
2. To the number obtained in the first step, add the last digit of the original number, multiplied by \(\displaystyle 5\).
A number is divisible by \(\displaystyle 7\) if and only if the sum obtained in the second step is divisible by \(\displaystyle 7\).
Explanation
Divisibility test by 7 for two-digit numbers
For a two-digit number, the test of divisibility by \(\displaystyle 7\) can be formulated as follows:
1. \(\displaystyle (\color(blue)X)(\color(red)Y)\rightarrow (\color(blue)X)\).
2. \(\displaystyle (\color(blue)X)+5\cdot(\color(red)Y)\).
The number \(\displaystyle (\color(blue)X)(\color(red)Y)\) is divisible by \(\displaystyle 7\) if and only if the number \(\displaystyle (\color(blue)X )+5\cdot(\color(red)Y)\) is divided by \(\displaystyle 7\).
The number \(\displaystyle 78\) is given. Let us carry out calculations in accordance with the rule described above.
1. We discard the last digit of the original number:
\(\displaystyle (\color(blue)7)(\color(red)8) \rightarrow (\color(blue)7)\).
2. Calculate:
\(\displaystyle (\color(blue)7)+5 \cdot (\color(red)8) = 47\).
The number \(\displaystyle 78\) is divisible by \(\displaystyle 7\) if and only if the number \(\displaystyle 47\) is divisible by \(\displaystyle 7\).
But since \(\displaystyle 47\) is not divisible by \(\displaystyle 7\), then \(\displaystyle 78\) is also not shared to \(\displaystyle 7\).
Answer: no, it is not divisible by \(\displaystyle 7\).
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.
From school curriculum many remember that there are signs of divisibility. This phrase refers to rules that allow you to quickly determine whether a number is a multiple of a given number without performing a direct arithmetic operation. This method is based on actions performed with part of the numbers from the entry in positional
Many people remember the simplest signs of divisibility from the school curriculum. For example, the fact that all numbers whose last digit is even are divisible by 2. This sign is the easiest to remember and apply in practice. If we talk about the method of dividing by 3, then the following rule applies for multi-digit numbers, which can be shown with this example. You need to find out whether 273 is a multiple of three. To do this, perform the following operation: 2+7+3=12. The resulting sum is divided by 3, therefore, 273 will be divided by 3 in such a way that the result will be an integer.
The signs of divisibility by 5 and 10 will be as follows. In the first case, the entry will end with the numbers 5 or 0, in the second case only with 0. In order to find out whether the dividend is a multiple of four, proceed as follows. It is necessary to isolate the last two digits. If these are two zeros or a number that is divisible by 4 without a remainder, then everything being divided will be a multiple of the divisor. It should be noted that the listed characteristics are used only in the decimal system. They are not used in other number methods. In such cases, their own rules are derived, which depend on the basis of the system.
The signs of division by 6 are as follows. 6 if it is a multiple of both 2 and 3. In order to determine whether a number is divisible by 7, you need to double the last digit in its notation. The resulting result is subtracted from the original number, which does not take into account the last digit. This rule can be seen in the following example. It is necessary to find out whether it is a multiple of 364. To do this, 4 is multiplied by 2, resulting in 8. Then the following action is performed: 36-8 = 28. The result obtained is a multiple of 7, and therefore the original number 364 can be divided by 7.
The signs of divisibility by 8 are as follows. If the last three digits in a number form a number that is a multiple of eight, then the number itself will be divisible by the given divisor.
You can find out whether a multi-digit number is divisible by 12 as follows. Using the divisibility criteria listed above, you need to find out whether the number is a multiple of 3 and 4. If they can simultaneously act as divisors for the number, then with a given dividend you can also perform the operation of division by 12. Similar rule also applies to other complex numbers, such as fifteen. In this case, the divisors should be 5 and 3. To find out whether a number is divisible by 14, you should see whether it is a multiple of 7 and 2. So, you can consider this in the following example. It is necessary to determine whether 658 can be divided by 14. The last digit in the entry is even, therefore, the number is a multiple of two. Next, we multiply 8 by 2, we get 16. From 65 we need to subtract 16. The result 49 is divided by 7, like the whole number. Therefore, 658 can be divided by 14.
If the last two digits in a given number are divisible by 25, then the whole number will be a multiple of this divisor. For multi-digit numbers, the sign of divisibility by 11 will sound as follows. It is necessary to find out whether a given divisor is a multiple of the difference between the sums of the digits that are in odd and even places in its notation.
It should be noted that the signs of divisibility of numbers and their knowledge very often greatly simplifies many problems that occur not only in mathematics, but also in everyday life. By being able to determine whether a number is a multiple of another, you can quickly complete various tasks. In addition, the use of these methods in mathematics classes will help develop students or schoolchildren and will contribute to the development of certain abilities.
