From the school curriculum, many remember that there are signs of divisibility. This phrase is understood as rules that allow you to quickly determine whether a number is a multiple of a given one, without performing a direct arithmetic operation. This method is based on actions performed with a part of the digits from the entry in the positional

Many people remember the simplest signs of divisibility from the school curriculum. For example, the fact that all numbers are divisible by 2, the last digit in the record of which is even. This feature is the easiest to remember and apply in practice. If we talk about the method of dividing by 3, then for multi-digit numbers the following rule applies, which can be shown in such an example. You need to find out if 273 is a multiple of three. To do this, perform the following operation: 2+7+3=12. The resulting sum is divisible by 3, therefore, 273 will be divisible by 3 in such a way that the result is 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 if the divisible is a multiple of four, proceed as follows. It is necessary to isolate the last two digits. If it is two zeros or a number that is divisible by 4 without a remainder, then everything divisible will be a multiple of the divisor. It should be noted that the listed signs are used only in the decimal system. They do not apply to other counting 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 entry. The result obtained is subtracted from the original number, in which the last digit is not taken into account. This rule can be seen in the following example. It is necessary to find out if 364 is a multiple. To do this, 4 is multiplied by 2, it turns out 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 if a multi-digit number is divisible by 12 as follows. Using the divisibility criteria listed above, you need to find out if the number is a multiple of 3 and 4. If they can simultaneously act as divisors for a number, then with a given divisible, you can also divide by 12. A similar rule applies to other complex numbers, for example, fifteen. In this case, the divisors should be 5 and 3. To find out if a number is divisible by 14, you should see if 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, you need to subtract 16. The result 49 is divisible by 7, like the whole number. Therefore, 658 can also be divided by 14.

If the last two digits in a given number are divisible by 25, then all of it 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 if the difference between the sums of digits that are in odd and even places in its record is a multiple of a given divisor.

It should be noted that the signs of divisibility of numbers and their knowledge very often greatly simplifies many tasks that are encountered not only in mathematics, but also in everyday life. Thanks to the ability to determine whether a number is a multiple of another, you can quickly perform various tasks. In addition, the use of these methods in mathematics classes will help develop students or schoolchildren, will contribute to the development of certain abilities.

Signs of divisibility of numbers on 2, 3, 4, 5, 6, 8, 9, 10, 11, 25 and other numbers it is useful to know for quickly solving problems on the Digital notation of a number. Instead of dividing one number by another, it is enough to check a number of signs, on the basis of which it is possible to unambiguously determine whether one number is divisible by another completely (whether it is a multiple) or not.

The main signs of divisibility

Let's bring main signs of divisibility of numbers:

• Sign of divisibility of a number by "2" The number is evenly divisible by 2 if the number is even (the last digit is 0, 2, 4, 6, or 8)
  Example: The number 1256 is a multiple of 2 because it ends in 6. And the number 49603 is not even divisible by 2 because it ends in 3.
• Sign of divisibility of a number by "3" A number is divisible by 3 if the sum of its digits is divisible by 3
  Example: The number 4761 is divisible by 3 because the sum of its digits is 18 and it is divisible by 3. And the number 143 is not a multiple of 3 because the sum of its digits is 8 and it is not divisible by 3.
• Sign of divisibility of a number by "4" A number is divisible by 4 if the last two digits of the number are zero or if the number made up of the last two digits is divisible by 4
  Example: The number 2344 is a multiple of 4 because 44 / 4 = 11. And the number 3951 is not divisible by 4 because 51 is not divisible by 4.
• Sign of divisibility of a number by "5" A number is divisible by 5 if the last digit of the number is 0 or 5
  Example: The number 5830 is divisible by 5 because it ends in 0. But the number 4921 is not divisible by 5 because it ends in 1.
• Sign of divisibility of a number by "6" A number is divisible by 6 if it is divisible by 2 and 3
  Example: The number 3504 is a multiple of 6 because it ends in 4 (the sign of divisibility by 2) and the sum of the digits of the number is 12 and it is divisible by 3 (the sign of divisibility by 3). And the number 5432 is not completely divisible by 6, although the number ends with 2 (the sign of divisibility by 2 is observed), but the sum of the digits is 14 and it is not completely divisible by 3.
• Sign of divisibility of a number by "8" A number is divisible by 8 if the last three digits of the number are zero or if the number made up of the last three digits of the number is divisible by 8
  Example: The number 93112 is divisible by 8 because 112 / 8 = 14. And the number 9212 is not a multiple of 8 because 212 is not divisible by 8.
• Sign of divisibility of a number by "9" A number is divisible by 9 if the sum of its digits is divisible by 9
  Example: The number 2916 is a multiple of 9, since the sum of the digits is 18 and it is divisible by 9. And the number 831 is not even divisible by 9, since the sum of the digits of the number is 12 and it is not divisible by 9.
• Sign of divisibility of a number by "10" A number is divisible by 10 if it ends in 0
  Example: The number 39590 is divisible by 10 because it ends in 0. And the number 5964 is not divisible by 10 because it doesn't end in 0.
• Sign of divisibility of a number by "11" A number is divisible by 11 if the sum of the digits in odd places is equal to the sum of the digits in even places or the sums must differ by 11
  Example: The number 3762 is divisible by 11 because 3 + 6 = 7 + 2 = 9. And the number 2374 is not divisible by 11 because 2 + 7 = 9 and 3 + 4 = 7.
• Sign of divisibility of a number by "25" A number is divisible by 25 if it ends in 00, 25, 50, or 75
  Example: The number 4950 is a multiple of 25 because it ends in 50. And 4935 is not divisible by 25 because it ends in 35.

Divisibility criteria for a composite number

To find out if a given number is divisible by a composite number, you need to decompose this composite number into relatively prime factors, whose divisibility criteria are known. Coprime numbers are numbers that have no common divisors other than 1. For example, a number is divisible by 15 if it is divisible by 3 and 5.

Consider another example of a compound divisor: a number is divisible by 18 if it is divisible by 2 and 9. In this case, you cannot decompose 18 into 3 and 6, since they are not coprime, since they have a common divisor of 3. We will verify this by example.

The number 456 is divisible by 3, since the sum of its digits is 15, and divisible by 6, since it is divisible by both 3 and 2. But if you manually divide 456 by 18, you get the remainder. If, for the number 456, we check the signs of divisibility by 2 and 9, it is immediately clear that it is divisible by 2, but not divisible by 9, since the sum of the digits of the number is 15 and it is not divisible by 9.

A series of articles on the signs of divisibility continues sign of divisibility by 3. This article first gives the formulation of the criterion for divisibility by 3, and gives examples of the application of this criterion in finding out which of the given integers are divisible by 3 and which are not. Further, the proof of the divisibility test by 3 is given. Approaches to establishing the divisibility by 3 of numbers given as the value of some expression are also considered.

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Sign of divisibility by 3, examples

Let's start with formulations of the test for divisibility by 3: an integer is divisible by 3 if the sum of its digits is divisible by 3 , if the sum of its digits is not divisible by 3 , then the number itself is not divisible by 3 .

From the above formulation it is clear that the sign of divisibility by 3 cannot be used without the ability to perform. Also, for the successful application of the sign of divisibility by 3, you need to know that of all the numbers 3, 6 and 9 are divisible by 3, and the numbers 1, 2, 4, 5, 7 and 8 are not divisible by 3.

Now we can consider the simplest examples of applying the test for divisibility by 3. Find out if the number −42 is divisible by 3. To do this, we calculate the sum of the digits of the number −42, it is equal to 4+2=6. Since 6 is divisible by 3, then, by virtue of the divisibility criterion by 3, it can be argued that the number −42 is also divisible by 3. But the positive integer 71 is not divisible by 3, since the sum of its digits is 7+1=8, and 8 is not divisible by 3.

Is 0 divisible by 3? To answer this question, the test for divisibility by 3 is not needed, here we need to recall the corresponding divisibility property, which states that zero is divisible by any integer. So 0 is divisible by 3 .

In some cases, to show that a given number has or does not have the ability to be divisible by 3, the test for divisibility by 3 has to be applied several times in a row. Let's take an example.

Example.

Show that the number 907444812 is divisible by 3.

Decision.

The sum of the digits of 907444812 is 9+0+7+4+4+4+8+1+2=39 . To find out if 39 is divisible by 3 , we calculate its sum of digits: 3+9=12 . And to find out if 12 is divisible by 3, we find the sum of the digits of the number 12, we have 1+2=3. Since we got the number 3, which is divisible by 3, then, due to the sign of divisibility by 3, the number 12 is divisible by 3. Therefore, 39 is divisible by 3, since the sum of its digits is 12, and 12 is divisible by 3. Finally, 907333812 is divisible by 3 because the sum of its digits is 39 and 39 is divisible by 3.

To consolidate the material, we will analyze the solution of another example.

Example.

Is the number −543205 divisible by 3?

Decision.

Let's calculate the sum of digits of this number: 5+4+3+2+0+5=19 . In turn, the sum of the digits of the number 19 is 1+9=10 , and the sum of the digits of the number 10 is 1+0=1 . Since we got the number 1, which is not divisible by 3, it follows from the criterion of divisibility by 3 that 10 is not divisible by 3. Therefore, 19 is not divisible by 3, because the sum of its digits is 10, and 10 is not divisible by 3. Therefore, the original number −543205 is not divisible by 3, since the sum of its digits, equal to 19, is not divisible by 3.

Answer:

No.

It is worth noting that the direct division of a given number by 3 also allows us to conclude whether the given number is divisible by 3 or not. By this we want to say that division should not be neglected in favor of the sign of divisibility by 3. In the last example, 543205 times 3 , we would make sure that 543205 is not even divisible by 3 , from which we could say that −543205 is not divisible by 3 either.

Proof of the test for divisibility by 3

The following representation of the number a will help us prove the sign of divisibility by 3. Any natural number a we can , after which it allows us to obtain a representation of the form , where a n , a n−1 , ..., a 0 are the digits from left to right in the notation of the number a . For clarity, we give an example of such a representation: 528=500+20+8=5 100+2 10+8 .

Now let's write a number of fairly obvious equalities: 10=9+1=3 3+1 , 100=99+1=33 3+1 , 1 000=999+1=333 3+1 and so on.

Substituting into equality a=a n 10 n +a n−1 10 n−1 +…+a 2 10 2 +a 1 10+a 0 instead of 10 , 100 , 1 000 and so on expressions 3 3+1 , 33 3+1 , 999+1=333 3+1 and so on, we get
.

And allow the resulting equality to be rewritten as follows:

Expression is the sum of the digits of a. Let's denote it for brevity and convenience by the letter A, that is, let's take . Then we get a representation of the number a of the form , which we will use in proving the test for divisibility by 3 .

Also, to prove the test for divisibility by 3, we need the following properties of divisibility:

• that an integer a is divisible by an integer b is necessary and sufficient that a is divisible by the modulus of b;
• if in the equality a=s+t all terms, except for some one, are divisible by some integer b, then this one term is also divisible by b.

Now we are fully prepared and can carry out proof of divisibility by 3, for convenience, we formulate this feature as a necessary and sufficient condition for divisibility by 3 .

Theorem.

For an integer a to be divisible by 3, it is necessary and sufficient that the sum of its digits is divisible by 3.

Proof.

For a=0 the theorem is obvious.

If a a is different from zero, then the modulus of a is a natural number, then the representation is possible, where is the sum of the digits of the number a.

Since the sum and product of integers is an integer, then is an integer, then by definition of divisibility, the product is divisible by 3 for any a 0 , a 1 , …, a n .

If the sum of the digits of the number a is divisible by 3, that is, A is divisible by 3, then, due to the divisibility property indicated before the theorem, it is divisible by 3, therefore, a is divisible by 3. This proves the sufficiency.

If a a is divisible by 3, then it is divisible by 3, then due to the same property of divisibility, the number A is divisible by 3, that is, the sum of the digits of the number a is divisible by 3. This proves the necessity.

Other cases of divisibility by 3

Sometimes integers are not specified explicitly, but as the value of some given value of the variable. For example, the value of an expression for some natural n is a natural number. It is clear that with this assignment of numbers, direct division by 3 will not help to establish their divisibility by 3, and the sign of divisibility by 3 will not always be able to be applied. Now we will consider several approaches to solving such problems.

The essence of these approaches is to represent the original expression as a product of several factors, and if at least one of the factors is divisible by 3, then, due to the corresponding property of divisibility, it will be possible to conclude that the entire product is divisible by 3.

Sometimes this approach allows you to implement. Let's consider an example solution.

Example.

Is the value of the expression divisible by 3 for any natural n ?

Decision.

The equality is obvious. Let's use Newton's binomial formula:

In the last expression, we can take 3 out of brackets, and we get . The resulting product is divisible by 3, since it contains a factor 3, and the value of the expression in brackets for natural n is a natural number. Therefore, is divisible by 3 for any natural n.

Answer:

Yes.

In many cases, proving divisibility by 3 allows . Let's analyze its application in solving an example.

Example.

Prove that for any natural n the value of the expression is divisible by 3 .

Decision.

For the proof, we use the method of mathematical induction.

At n=1 the value of the expression is , and 6 is divisible by 3 .

Suppose the value of the expression is divisible by 3 when n=k , that is, divisible by 3 .

Taking into account that it is divisible by 3 , we will show that the value of the expression for n=k+1 is divisible by 3 , that is, we will show that is divisible by 3.

Mathematics in grade 6 begins with the study of the concept of divisibility and signs of divisibility. Often limited to signs of divisibility by such numbers:

• On the 2 : last digit must be 0, 2, 4, 6 or 8;
• On the 3 : the sum of the digits of the number must be divisible by 3;
• On the 4 : the number formed by the last two digits must be divisible by 4;
• On the 5 : last digit must be 0 or 5;
• On the 6 : the number must have signs of divisibility by 2 and 3;
• Sign of divisibility by 7 often skipped;
• Rarely do they also talk about the test for divisibility into 8 , although it is similar to the signs of divisibility by 2 and 4. For a number to be divisible by 8, it is necessary and sufficient that the three-digit ending be divisible by 8.
• Sign of divisibility by 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.
• Sign of divisibility by 10 , probably the simplest: the number must end in zero.
• Sometimes sixth graders are also told about the sign of divisibility into 11 . You need to add the digits of the number in even places, subtract the numbers in odd places from the result. If the result is divisible by 11, then the number itself is divisible by 11.

Let us now return to the sign of divisibility by 7. If they talk about it, it is combined with the sign of divisibility by 13 and it is advised to use it that way.

We 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 b 11).

This method is based, as well as a number of mathematical tricks, on the fact that 7x11x13 \u003d 1001. However, what to do with three-digit numbers, for which the question of divisibility, sometimes, cannot be solved without division itself.

Using the universal divisibility test, one can build relatively simple algorithms for determining whether a number is divisible by 7 and other "inconvenient" numbers.

Improved test for divisibility by 7
To check if 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 performed multiple times.
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 didn't notice this, we could take 1 more step: 6-3-3 = 0, and 0 is definitely divisible by 7).

So the number 65835 is also divisible by 7.

Based on the universal divisibility criterion, it is possible to improve the divisibility criteria 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, so the number is divisible by 4.

Example 4
Is the number 134 divisible by 4?
3+4/2 = 5, odd number, so 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, so 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, so 1984 is divisible by 8.

Sign of divisibility by 12 is the union of the signs of divisibility by 3 and by 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 of pq, such that gcd(p,q)=1), one must be divisible by both p and q at the same time.

However, be careful! In order for the composite signs of divisibility to work, the factors of the number must be precisely 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 if 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 either.

Example 8
Is 715 divisible by 13?
71+4*5 = 91
9+4*1 = 13
13 is divisible by 13, so 715 is also 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 criteria for divisibility by 12. We check the divisibility by coprime factors of these numbers.

• For 14: for 2 and for 7;
• For 15: by 3 and by 5;
• For 18: 2 and 9;
• For 21: on 3 and on 7;
• For 20: by 4 and by 5 (or, in other words, the last digit must be zero, and the penultimate one must be even);
• For 24: 3 and 8;
• For 26: 2 and 13;
• For 28: 4 and 7.

Improved test for divisibility by 16.
Instead of checking to see if the 4-digit ending is divisible by 16, you can add the units digit with ten times the tens digit, quadruple the hundreds digit, and
eight times the thousand digit, and check if the result is divisible by 16.

Example 9
Is 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, so 1984 is not divisible by 16 either.

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, so 1526 is also divisible by 16.

Improved test for divisibility by 17.
To check if a number is divisible by 17, you need to discard the last digit from the number and subtract this figure 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, so 59772 is also divisible by 17.

Example 12
Is 4913 divisible by 17?
491-5*3 = 476
47-5*6 = 17
17 is divisible by 17, so 4913 is also divisible by 17.

Improved test for divisibility by 19.
To check if 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, so 9044 is also divisible by 19.

Improved test for divisibility by 23.
To check if 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 see that 253 is 23,

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

Sign of divisibility by 2

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

Sign of divisibility by 3

This sign of divisibility has completely different rules: if the sum of the digits of a number is divisible by 3, then the number is 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.
So, in order to understand whether a number is divisible by 3, you just need to add together the numbers that make it up.
It looks like this: 3987 and 141 are divided by 3, because in the first case 3+9+8+7= 27 (27:3=9 - divisible without remainder by 3), and in the second 1+4+1= 6 (6:3=2 - also divisible by 3 without remainder).
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 can be divided by 3 without a remainder).

Divisibility by 4 sign

This test of divisibility 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 divisible by 4, because in the first case the number ends in 00 , and in the second 64 , which in turn is divisible by 4 without a remainder (64:4=16)
Numbers 3 57 and 8 86 are not divisible by 4 because neither 57 neither 86 are not divisible by 4, and therefore do not correspond to this criterion of divisibility.

Sign of divisibility by 5

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

Sign of divisibility by 6

Before us is a composite number 6, which is the product of the numbers 2 and 3. Therefore, the sign of divisibility by 6 is also composite: in order for a number to be divisible by 6, it must correspond to two signs of divisibility at the same time: the sign of divisibility by 2 and the sign of divisibility by 3. At the same time, note that such a composite number as 4 has an individual sign of divisibility, because it is a product of the number 2 by itself. But back to the test for divisibility by 6.
The numbers 138 and 474 are even and correspond to the signs 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, and therefore do not correspond to the criterion of divisibility by 2, and therefore do not correspond to the criterion of divisibility by 6.

Sign of divisibility by 7

This divisibility criterion is more complicated: a number is divisible by 7 if the result of subtracting the last digit from the number of tens of this number is divisible by 7 or equals 0.
It sounds rather confusing, but in practice it is simple. See for yourself: number 95 9 is divisible by 7 because 95 -2*9=95-18=77, 77:7=11 (77 is divisible by 7 without a remainder). Moreover, if there are difficulties with the number obtained during the transformations (due to its size, it is difficult to understand whether it is divisible by 7 or not, then this procedure can be continued as many times as you see fit).
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 is difficult for us to understand whether 457 8 by 7, so let's repeat the process: 457 -2*8=457-16=441. And again we will use the sign of divisibility, since we still have a three-digit number in front of us 44 1. So, 44 -2*1=44-2=42, 42:7=6, i.e. 42 is divisible by 7 without a remainder, which means that 45801 is also divisible by 7.
And here are 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).

Sign of divisibility by 8

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

Sign of 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.
For example: 3987 and 144 are divisible by 9 because in the first case 3+9+8+7= 27 (27:9=3 - divisible without remainder by 9), and in the second 1+4+4= 9 (9:9=1 - also divisible without remainder by 9).
But the numbers: 235 and 141 are not divisible by 9, because 2+3+5= 10 and 1+4+1= 6 (and we know that neither 10 nor 6 can be divided by 9 without a remainder).

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

I combined these divisibility criteria because they can be described in the same way: a number is divisible 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 . all of which are divisible by 1 0 ; 46400 and 867 000 are also divisible by 1 00 ; and only one of them - 867 000 divisible by 1 000 .
Any numbers that end in zeroes less than a bit unit are not divisible by that bit unit, such as 600 30 and 7 93 do not share 1 00 .

Sign of divisibility by 11

In order to find out if a number is divisible by 11, you need to get the difference between the sums of even and odd digits of this number. If this difference is equal to 0 or 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 because ( 9 +9 )-(2+1+4)=18-7=11.
And here is 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.

Sign of divisibility by 12

The number 12 is composite. Its sign of divisibility is the correspondence to the signs of divisibility by 3 and by 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 divisible by 4) and the signs of divisibility by 3 (the sum of the digits and the first and second numbers are divisible by 3), and therefore, 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 both signs at the same time.

Sign of divisibility by 13

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

Divisibility of numbers

As can be seen from the above, it can be assumed that any of the natural numbers can be matched with its own individual sign of divisibility or a "composite" sign if the number is a multiple of several different numbers. But as practice shows, basically the larger the number, the more complex its feature. Perhaps the time spent on checking the divisibility criterion may be equal to or greater than the division itself. That is why we usually use the simplest of divisibility tests.