How to find out if a number is divisible. Signs of divisibility, whether a number is divisible

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.

Page navigation.

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.

Solution.

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?

Solution.

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 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 is divisible by 3, then it is divisible by 3, then due to the same divisibility property, 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 ?

Solution.

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 .

Solution.

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.

Sign of divisibility by 2

A number is divisible by two if it last digit is even or zero. In other cases, it does not share.

For example:

Number 52 73 8 is divisible by 2 because the last digit of 8 is even.
7 691 is not divisible by 2, so 1 is an odd digit.
1 250 is divisible by 2 since the last digit is zero.

Signs of divisibility by 3

Only those numbers are divisible by 3 for which the sum of the digits is divisible for 3

For example:

The number 17835 is divisible by 3 because the sum of its digits

\[ 1 + 7 + 8 + 3 + 5 = 24 \]

is divisible by 3.

Divisibility by 4 sign

A number is divisible by 4 if its last two digits are zeros or form a number divisible by by 4 . In other cases, it does not share.

Examples:

31,700 is divisible by 4 because it ends in two zeros.
4 215 634 is not divisible by 4 because the last two digits give 34, which is not divisible by 4.
16608 is divisible by 4 because the last two digits of 08 are 8 divisible by 4 .

Sign of divisibility by 5

Numbers are divisible by 5 the last digit of which 0 or 5 . Others don't share.

Example:

240 is divisible by 5 (the last digit is 0).
554 is not divisible by 5 (the last digit is 4).

Sign of divisibility by 6

A number is divisible by 6 if it is divides at the same time for both 2 and 3. Otherwise, it does not share.

For example:

126 is divisible by 6 because it is divisible by 2 and 3.

Sign of divisibility by 8

Similar to the test for divisibility by 4. The number is divisible by 8 if its last three digits are zeros or form a number that is divisible by 8 . In other cases, it does not share.

Examples:

125,000 is divisible by 8 (three zeros at the end).
170004 is not divisible by 8 (the last three digits give 4, which is not divisible by 8).
111120 is divisible by 8 (the last three digits give the number 120 divisible by 8).

Notes. You can specify similar signs for dividing by 16, 32, 64, etc., but they are of no practical importance.

Sign of divisibility by 9

Only those numbers are divisible by 9 for which the sum of the digits is divisible at 9 .

Examples:

The number 106499 is not divisible by 9 because the sum of its digits (29) is not divisible by 9. The number 52632 is divisible by 9 because the sum of its digits (18) is divisible by 9.

Signs of divisibility by 10, 100 and 1000

Only those numbers are divisible by 10 the last digit is zero, by 100 - only those numbers whose last two digits are zeros, by 1000 - only those whose last three digits are zeros.

Examples:

8200 is divisible by 10 and by 100 .
542,000 is divisible by 10, 100, 1000.

Sign of divisibility by 11

Only those numbers are divisible by 11 for which the sum of the digits occupying odd places is either equal to the sum of the digits occupying even places, or differs from it by a number divisible by 11.

Examples:

The number 103 785 is divisible by 11, since the sum of the digits in odd places is

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 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;
• 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 ones 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,

Definition 1. Let the number a 1) there is a product of two numbers b And q So a=bq. Then a is called a multiple b.

1) In this article, the word number will mean an integer.

You can also say a divided by b, or b there is a divisor a, or b divides a, or b enters as a factor in a.

Definition 1 implies the following assertions:

Statement1. If a-multiple b, b-multiple c, That a multiple c.

Really. Because

Where m And n some numbers,

Hence a divided by c.

If in a series of numbers, each is divisible by the next one, then each number is a multiple of all subsequent numbers.

Statement 2. If numbers a And b- multiples c, then their sum and difference are also multiples c.

Really. Because

a+b=mc+nc=(m+n)c,

a−b=mc−nc=(m−n)c.

Hence a+b divided by c And a−b divided by c .

Signs of divisibility

We derive a general formula for determining the sign of divisibility of numbers by some natural number m, which is called Pascal's divisibility test.

Find the remainder of division by m next sequence. Let the remainder of dividing 10 by m will r 1, 10· r 1 on m will r 2 , etc. Then you can write:

Let us prove that the remainder of the division of the number A on m equal to the remainder of dividing the number

(3)

As you know, if two numbers, when divided by some number m give the same remainder, then the difference is divisible by m without a trace.

Consider the difference A−A"

(6)

(7)

Each term on the right side of (5) is divisible by m hence the left side of the equation is also divisible by m. Arguing similarly, we get - the right side of (6) is divided by m, therefore, the left side of (6) is also divisible by m, the right side of (7) is divisible by m, therefore, the left side of (7) is also divisible by m. We found that the right side of equation (4) is divisible by m. Hence A And A" have the same remainder when divided by m. In this case, they say that A And A" equidistant or comparable in modulus m.

Thus, if A" divided by m m) , That A also divided into m(has zero remainder when divided by m). We have shown that in order to determine the divisibility A it is possible to determine the divisibility of a simpler number A".

Based on expression (3), it is possible to obtain signs of divisibility for specific numbers.

Signs of divisibility of numbers 2, 3, 4, 5, 6, 7, 8, 9, 10

Sign of divisibility by 2.

Following procedure (1) for m=2, we get:

All remainders after dividing by 2 are zero. Then, from equation (3) we have

All remainders after dividing by 3 are equal to 1. Then, from equation (3) we have

All remainders from division by 4 except the first one are equal to 0. Then, from equation (3) we have

All remainders are zero. Then, from equation (3) we have

All remainders are equal to 4. Then, from equation (3) we have

Therefore, the number is divisible by 6 if and only if the quadruple number of tens, added to the number of ones, is divisible by 6. That is, we discard the right digit from the number, then sum the resulting number with 4 and add the discarded number. If the given number is divisible by 6, then the original number is divisible by 6.

Example. 2742 is divisible by 6 because 274*4+2=1098, 1098=109*4+8=444, 444=44*4+4=180 is divisible by 6.

A simpler criterion for divisibility. A number is divisible by 6 if it is divisible by 2 and 3 (i.e. if it even number and if the sum of the digits is divisible by 3). The number 2742 is divisible by 6 because the number is even and 2+7+4+2=15 is divisible by 3.

Sign of divisibility by 7.

Following procedure (1) for m=7, we get:

All residues are different and are repeated after 7 steps. Then, from equation (3) we have

All remainders are all zero, except for the first two. Then, from equation (3) we have

All remainders after division by 9 are equal to 1. Then, from equation (3) we have

All remainders after dividing by 10 are 0. Then, from equation (3) we have

Therefore, a number is divisible by 10 if and only if the last digit is divisible by 10 (that is, the last digit is zero).

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 expand this composite number on mutually prime factors , whose divisibility criteria are known. Mutually prime 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.