Division natural numbers, especially polysemantic ones, are conveniently carried out using a special method, which is called division by a column (in a column). You can also find the name corner division. Let us immediately note that the column can be used to both divide natural numbers without a remainder and divide natural numbers with a remainder.

In this article we will look at how long division is performed. Here we will talk about recording rules and all intermediate calculations. First, let's focus on dividing a multi-digit natural number by a single-digit number with a column. After this, we will focus on cases when both the dividend and the divisor are multi-valued natural numbers. The entire theory of this article is provided typical examples dividing natural numbers with a column with detailed explanations of the solution and illustrations.

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Rules for recording when dividing by a column

Let's start by studying the rules for writing the dividend, divisor, all intermediate calculations and results when dividing natural numbers by a column. Let's say right away that it is most convenient to do column division in writing on paper with a checkered line - this way there is less chance of straying from the desired row and column.

First, the dividend and divisor are written in one line from left to right, after which a symbol of the form is drawn between the written numbers. For example, if the dividend is the number 6 105 and the divisor is 5 5, then their correct notation when dividing into a column will be as follows:

Look at the following diagram to illustrate where to write the dividend, divisor, quotient, remainder, and intermediate calculations in long division.

From the above diagram it is clear that the required quotient (or incomplete quotient when dividing with a remainder) will be written below the divisor under the horizontal line. And intermediate calculations will be carried out below the dividend, and you need to take care in advance about the availability of space on the page. In this case, one should be guided by the rule: what more difference in the number of digits in the dividend and divisor entries, the more space is required. For example, when dividing by a column the natural number 614,808 by 51,234 (614,808 is a six-digit number, 51,234 is a five-digit number, the difference in the number of characters in the records is 6−5 = 1), intermediate calculations will require less space than when dividing the numbers 8 058 and 4 (here the difference in the number of characters is 4−1=3). To confirm our words, we present complete records of division by a column of these natural numbers:

Now you can proceed directly to the process of dividing natural numbers by a column.

Column division of a natural number by a single-digit natural number, column division algorithm

It is clear that dividing one single-digit natural number by another is quite simple, and there is no reason to divide these numbers into a column. However, it will be helpful to practice your initial long division skills with these simple examples.

Example.

Let us need to divide with a column of 8 by 2.

Solution.

Of course, we can perform division using the multiplication table, and immediately write down the answer 8:2=4.

But we are interested in how to divide these numbers with a column.

First, we write down the dividend 8 and the divisor 2 as required by the method:

Now we begin to find out how many times the divisor is contained in the dividend. To do this, we sequentially multiply the divisor by the numbers 0, 1, 2, 3, ... until the result is a number equal to the dividend (or a number greater than the dividend, if there is a division with a remainder). If we get a number equal to the dividend, then we immediately write it under the dividend, and in the place of the quotient we write the number by which we multiplied the divisor. If we get a number greater than the dividend, then under the divisor we write the number calculated at the penultimate step, and in place of the incomplete quotient we write the number by which the divisor was multiplied at the penultimate step.

Let's go: 2·0=0 ; 2·1=2 ; 2·2=4 ; 2·3=6 ; 2·4=8. We have received a number equal to the dividend, so we write it under the dividend, and in place of the quotient we write the number 4. In this case, the entry will accept next view:

The final stage of dividing single-digit natural numbers with a column remains. Under the number written under the dividend, you need to draw a horizontal line, and subtract the numbers above this line in the same way as is done when subtracting natural numbers in a column. The number resulting from the subtraction will be the remainder of the division. If it is equal to zero, then the original numbers are divided without a remainder.

In our example we get

Now we have before us a completed recording of the column division of the number 8 by 2. We see that the quotient of 8:2 is 4 (and the remainder is 0).

Answer:

8:2=4 .

Now let's look at how a column divides single-digit natural numbers with a remainder.

Example.

Divide 7 by 3 using a column.

Solution.

At the initial stage, the entry looks like this:

We begin to find out how many times the dividend contains the divisor. We will multiply 3 by 0, 1, 2, 3, etc. until we get a number equal to or greater than the dividend 7. We get 3·0=0<7 ; 3·1=3<7 ; 3·2=6<7 ; 3·3=9>7 (if necessary, refer to the article comparing natural numbers). Under the dividend we write the number 6 (it was obtained at the penultimate step), and in place of the incomplete quotient we write the number 2 (the multiplication was carried out by it at the penultimate step).

It remains to carry out the subtraction, and the division by a column of single-digit natural numbers 7 and 3 will be completed.

Thus, the partial quotient is 2 and the remainder is 1.

Answer:

7:3=2 (rest. 1) .

Now you can move on to dividing multi-digit natural numbers by columns into single-digit natural numbers.

Now we'll figure it out long division algorithm. At each stage, we will present the results obtained by dividing the multi-digit natural number 140,288 by the single-digit natural number 4. This example was not chosen by chance, since when solving it we will encounter all possible nuances and will be able to analyze them in detail.

First we look at the first digit on the left in the dividend notation. If the number defined by this figure is greater than the divisor, then in the next paragraph we have to work with this number. If this number is less than the divisor, then we need to add to the consideration the next digit on the left in the notation of the dividend, and continue to work with the number determined by the two digits under consideration. For convenience, we highlight in our notation the number with which we will work.

The first digit from the left in the notation of the dividend 140288 is the digit 1. The number 1 is less than the divisor 4, so we also look at the next digit on the left in the notation of the dividend. At the same time, we see the number 14, with which we have to work further. We highlight this number in the notation of the dividend.

The following steps from the second to the fourth are repeated cyclically until the division of natural numbers by a column is completed.

Now we need to determine how many times the divisor is contained in the number we are working with (for convenience, let's denote this number as x). To do this, we sequentially multiply the divisor by 0, 1, 2, 3, ... until we get the number x or a number greater than x. When the number x is obtained, we write it under the highlighted number according to the recording rules used when subtracting natural numbers in a column. The number by which the multiplication was carried out is written in place of the quotient during the first pass of the algorithm (in subsequent passes of 2-4 points of the algorithm, this number is written to the right of the numbers already there). When we get a number that is greater than the number x, then under the highlighted number we write the number obtained at the penultimate step, and in place of the quotient (or to the right of the numbers already there) we write the number by which the multiplication was carried out at the penultimate step. (We carried out similar actions in the two examples discussed above).

Multiply the divisor 4 by the numbers 0, 1, 2, ... until we get a number that is equal to 14 or greater than 14. We have 4·0=0<14 , 4·1=4<14 , 4·2=8<14 , 4·3=12<14 , 4·4=16>14 . Since at the last step we received the number 16, which is greater than 14, then under the highlighted number we write the number 12, which was obtained at the penultimate step, and in place of the quotient we write the number 3, since in the penultimate point the multiplication was carried out precisely by it.


At this stage, from the selected number, subtract the number located under it using a column. The result of the subtraction is written under the horizontal line. However, if the result of the subtraction is zero, then it does not need to be written down (unless the subtraction at that point is the very last action that completely completes the process of long division). Here, for your own control, it would not be amiss to compare the result of the subtraction with the divisor and make sure that it is less than the divisor. Otherwise, a mistake was made somewhere.

We need to subtract the number 12 from the number 14 with a column (for the correctness of the recording, we must remember to put a minus sign to the left of the numbers being subtracted). After completing this action, the number 2 appeared under the horizontal line. Now we check our calculations by comparing the resulting number with the divisor. Since the number 2 is less than the divisor 4, you can safely move on to the next point.


Now, under the horizontal line to the right of the numbers located there (or to the right of the place where we did not write down the zero), we write down the number located in the same column in the notation of the dividend. If there are no numbers in the record of the dividend in this column, then the division by column ends there. After this, we select the number formed under the horizontal line, accept it as a working number, and repeat points 2 to 4 of the algorithm with it.

Under the horizontal line to the right of the number 2 already there, we write down the number 0, since it is the number 0 that is in the record of the dividend 140,288 in this column. Thus, the number 20 is formed under the horizontal line.


We select this number 20, take it as a working number, and repeat with it the actions of the second, third and fourth points of the algorithm.

Multiply the divisor 4 by 0, 1, 2, ... until we get the number 20 or a number that is greater than 20. We have 4·0=0<20 , 4·1=4<20 , 4·2=8<20 , 4·3=12<20 , 4·4=16<20 , 4·5=20 . Так как мы получили число, равное числу 20 , то записываем его под отмеченным числом, а на месте частного, справа от уже имеющегося там числа 3 записываем число 5 (на него производилось умножение).


We carry out the subtraction in a column. Since we are subtracting equal natural numbers, then by virtue of the property of subtracting equal natural numbers, the result is zero. We do not write down the zero (since this is not the final stage of division with a column), but we remember the place where we could write it (for convenience, we will mark this place with a black rectangle).


Under the horizontal line to the right of the remembered place we write down the number 2, since it is precisely it that is in the record of the dividend 140,288 in this column. Thus, under the horizontal line we have the number 2.


We take the number 2 as the working number, mark it, and we will once again have to perform the actions of 2-4 points of the algorithm.

We multiply the divisor by 0, 1, 2, and so on, and compare the resulting numbers with the marked number 2. We have 4·0=0<2 , 4·1=4>2. Therefore, under the marked number we write the number 0 (it was obtained at the penultimate step), and in the place of the quotient to the right of the number already there we write the number 0 (we multiplied by 0 at the penultimate step).


We perform the subtraction in a column, we get the number 2 under the horizontal line. We check ourselves by comparing the resulting number with the divisor 4. Since 2<4 , то можно спокойно двигаться дальше.


Under the horizontal line to the right of the number 2, add the number 8 (since it is in this column in the entry for the dividend 140 288). Thus, the number 28 appears under the horizontal line.


We take this number as a working number, mark it, and repeat steps 2-4.

There shouldn't be any problems here if you have been careful up to now. Having completed all the necessary steps, the following result is obtained.

All that remains is to carry out the steps from points 2, 3, 4 one last time (we leave this to you), after which you will get a complete picture of dividing the natural numbers 140,288 and 4 into a column:

Please note that the number 0 is written in the very bottom line. If this was not the last step of division by a column (that is, if in the record of the dividend there were numbers left in the columns on the right), then we would not write this zero.

Thus, looking at the completed record of dividing the multi-digit natural number 140,288 by the single-digit natural number 4, we see that the quotient is the number 35,072 (and the remainder of the division is zero, it is in the very bottom line).

Of course, when dividing natural numbers by a column, you will not describe all your actions in such detail. Your solutions will look something like the following examples.

Example.

Perform long division if the dividend is 7 136 and the divisor is a single-digit natural number 9.

Solution.

At the first step of the algorithm for dividing natural numbers by columns, we get a record of the form

After performing the actions from the second, third and fourth points of the algorithm, the column division record will take the form

Repeating the cycle, we will have

One more pass will give us a complete picture of the column division of the natural numbers 7,136 and 9

Thus, the partial quotient is 792, and the remainder is 8.

Answer:

7 136:9=792 (remaining 8) .

And this example demonstrates what long division should look like.

Example.

Divide the natural number 7,042,035 by the single-digit natural number 7.

Solution.

The most convenient way to do division is by column.

Answer:

7 042 035:7=1 006 005 .

Column division of multi-digit natural numbers

We hasten to please you: if you have thoroughly mastered the column division algorithm from the previous paragraph of this article, then you almost already know how to perform column division of multi-digit natural numbers. This is true, since stages 2 to 4 of the algorithm remain unchanged, and only minor changes appear in the first point.

At the first stage of dividing multi-digit natural numbers into a column, you need to look not at the first digit on the left in the notation of the dividend, but at the number of them equal to the number of digits contained in the notation of the divisor. If the number defined by these numbers is greater than the divisor, then in the next paragraph we have to work with this number. If this number is less than the divisor, then we need to add to the consideration the next digit on the left in the notation of the dividend. After this, the actions specified in paragraphs 2, 3 and 4 of the algorithm are performed until the final result is obtained.

All that remains is to see the application of the column division algorithm for multi-valued natural numbers in practice when solving examples.

Example.

Let's perform column division of multi-digit natural numbers 5,562 and 206.

Solution.

Since the divisor 206 contains 3 digits, we look at the first 3 digits on the left in the dividend 5,562. These numbers correspond to the number 556. Since 556 is greater than the divisor 206, we take the number 556 as a working number, select it, and move on to the next stage of the algorithm.

Now we multiply the divisor 206 by the numbers 0, 1, 2, 3, ... until we get a number that is either equal to 556 or greater than 556. We have (if multiplication is difficult, then it is better to multiply natural numbers in a column): 206 0 = 0<556 , 206·1=206<556 , 206·2=412<556 , 206·3=618>556. Since we received a number that is greater than the number 556, then under the highlighted number we write the number 412 (it was obtained at the penultimate step), and in place of the quotient we write the number 2 (since we multiplied by it at the penultimate step). The column division entry takes the following form:

We perform column subtraction. We get the difference 144, this number is less than the divisor, so you can safely continue performing the required actions.

Under the horizontal line to the right of the number there we write the number 2, since it is in the record of the dividend 5562 in this column:

Now we work with the number 1,442, select it, and go through steps two through four again.

Multiply the divisor 206 by 0, 1, 2, 3, ... until you get the number 1442 or a number that is greater than 1442. Let's go: 206·0=0<1 442 , 206·1=206<1 442 , 206·2=412<1 332 , 206·3=618<1 442 , 206·4=824<1 442 , 206·5=1 030<1 442 , 206·6=1 236<1 442 , 206·7=1 442 . Таким образом, под отмеченным числом записываем 1 442 , а на месте частного правее уже имеющегося там числа записываем 7 :

We carry out the subtraction in a column, we get zero, but we don’t write it down right away, we just remember its position, because we don’t know whether the division ends here, or whether we’ll have to repeat the steps of the algorithm again:

Now we see that we cannot write any number under the horizontal line to the right of the remembered position, since there are no digits in the record of the dividend in this column. Therefore, this completes the division by column, and we complete the entry:

• Mathematics. Any textbooks for 1st, 2nd, 3rd, 4th grades of general education institutions.
• Mathematics. Any textbooks for 5th grade of general education institutions.

With this math program you can divide polynomials by column.
The program for dividing a polynomial by a polynomial does not just give the answer to the problem, it provides a detailed solution with explanations, i.e. displays the solution process to test knowledge in mathematics and/or algebra.

This program can be useful for high school students in general education schools when preparing for tests and exams, when testing knowledge before the Unified State Exam, and for parents to control the solution of many problems in mathematics and algebra. Or maybe it’s too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get your math or algebra homework done as quickly as possible? In this case, you can also use our programs with detailed solutions.

In this way, you can conduct your own training and/or training of your younger brothers or sisters, while the level of education in the field of solving problems increases.

If you need or simplify polynomial or multiply polynomials, then for this we have a separate program Simplification (multiplication) of a polynomial

First polynomial (divisible - what we divide):

Second polynomial (divisor - what we divide by):

Divide polynomials

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A little theory.

Dividing a polynomial into a polynomial (binomial) by a column (corner)

In algebra dividing polynomials with a column (corner)- an algorithm for dividing a polynomial f(x) by a polynomial (binomial) g(x), the degree of which is less than or equal to the degree of the polynomial f(x).

The polynomial-by-polynomial division algorithm is a generalized form of column division of numbers that can be easily implemented by hand.

For any polynomials \(f(x) \) and \(g(x) \), \(g(x) \neq 0 \), there are unique polynomials \(q(x) \) and \(r(x ) \), such that
\(\frac(f(x))(g(x)) = q(x)+\frac(r(x))(g(x)) \)
and \(r(x)\) has a lower degree than \(g(x)\).

The goal of the algorithm for dividing polynomials into a column (corner) is to find the quotient \(q(x) \) and the remainder \(r(x) \) for a given dividend \(f(x) \) and non-zero divisor \(g(x) \)

Example

Let's divide one polynomial by another polynomial (binomial) using a column (corner):
\(\large \frac(x^3-12x^2-42)(x-3) \)

The quotient and remainder of these polynomials can be found by performing the following steps:
1. Divide the first element of the dividend by the highest element of the divisor, place the result under the line \((x^3/x = x^2)\)

\(x\) \(-3 \) \(x^2\)

3. Subtract the polynomial obtained after multiplication from the dividend, write the result under the line \((x^3-12x^2+0x-42-(x^3-3x^2)=-9x^2+0x-42) \)

\(x^3\) \(-12x^2\) \(+0x\) \(-42 \) \(x^3\) \(-3x^2\) \(-9x^2\) \(+0x\) \(-42 \)

\(x\) \(-3 \) \(x^2\)

4. Repeat the previous 3 steps, using the polynomial written under the line as the dividend.

\(x^3\) \(-12x^2\) \(+0x\) \(-42 \) \(x^3\) \(-3x^2\) \(-9x^2\) \(+0x\) \(-42 \) \(-9x^2\) \(+27x\) \(-27x\) \(-42 \)

\(x\) \(-3 \) \(x^2\) \(-9x\)

5. Repeat step 4.

\(x^3\) \(-12x^2\) \(+0x\) \(-42 \) \(x^3\) \(-3x^2\) \(-9x^2\) \(+0x\) \(-42 \) \(-9x^2\) \(+27x\) \(-27x\) \(-42 \) \(-27x\) \(+81 \) \(-123 \)

\(x\) \(-3 \) \(x^2\) \(-9x\) \(-27 \)

6. End of the algorithm.
Thus, the polynomial \(q(x)=x^2-9x-27\) is the quotient of the division of polynomials, and \(r(x)=-123\) is the remainder of the division of polynomials.

The result of dividing polynomials can be written in the form of two equalities:
\(x^3-12x^2-42 = (x-3)(x^2-9x-27)-123\)
or
\(\large(\frac(x^3-12x^2-42)(x-3)) = x^2-9x-27 + \large(\frac(-123)(x-3)) \)

Children in grades 2-3 are learning a new mathematical operation - division. It is not easy for a student to understand the essence of this mathematical operation, so he needs the help of his parents. Parents need to understand exactly how to present new information to their child. TOP 10 examples will tell parents how to teach children how to divide numbers in a column.

Learning long division in the form of a game

Children get tired at school, they get tired of textbooks. Therefore, parents need to give up textbooks. Present information in the form of a fun game.

You can set tasks this way:

1 Organize a place for your child to learn through play. Place his toys in a circle, and give the child pears or candy. Have the student divide 4 candies between 2 or 3 dolls. To achieve understanding on the part of the child, gradually increase the number of candies to 8 and 10. Even if the baby takes a long time to act, do not put pressure or yell at him. You will need patience. If your child does something wrong, correct him calmly. Then, after he completes the first action of dividing the candies between the participants in the game, he will ask him to calculate how many candies went to each toy. Now the conclusion. If there were 8 candies and 4 toys, then each got 2 candies. Let your child understand that sharing means distributing an equal amount of candy to all toys.

2 You can teach math operations using numbers. Let the student understand that numbers can be classified as pears or candy. Say that the number of pears to be divided is the dividend. And the number of toys that contain candy is the divisor.

3 Give your child 6 pears. Give him a task: to divide the number of pears between grandfather, dog and dad. Then ask him to divide 6 pears between grandpa and dad. Explain to your child the reason why the division results were different.

4 Teach your student about division with a remainder. Give your child 5 candies and ask him to distribute them equally between the cat and dad. The child will have 1 candy left. Tell your child why it happened this way. This mathematical operation should be considered separately, as it can cause difficulties.

Playful learning can help your child quickly understand the whole process of dividing numbers. He will be able to learn that the largest number is divided by the smallest or vice versa. That is, the largest number is candy, and the smallest number is the participants. In column 1 the number will be the number of candies, and 2 will be the number of participants.

Do not overload your child with new knowledge. You need to learn gradually. You need to move on to new material when the previous material is consolidated.

Learning long division using the multiplication table

Students up to 5th grade will be able to understand division more quickly if they have a good understanding of multiplication.

Parents need to explain that division is similar to the multiplication table. Only the actions are opposite. For clarity, we need to give an example:

• Tell the student to freely multiply the values ​​of 6 and 5. The answer is 30.
• Tell the student that the number 30 is the result of a mathematical operation with two numbers: 6 and 5. Namely, the result of multiplication.
• Divide 30 by 6. The result of the mathematical operation is 5. The student will be able to see that division is the same as multiplication, but in reverse.

You can use the multiplication table to illustrate division if the child has mastered it well.

Learning long division in a notebook

Learning should begin when the student understands the material about division in practice, using games and multiplication tables.

You need to start dividing in this way, using simple examples. So, divide 105 by 5.

The mathematical operation needs to be explained in detail:

• Write an example in your notebook: 105 divided by 5.
• Write this down as you would for long division.
• Explain that 105 is the dividend and 5 is the divisor.
• With a student, identify 1 number that can be divided. The value of the dividend is 1, this figure is not divisible by 5. But the second number is 0. The result is 10, this value can be divided in this example. The number 5 is included in the number 10 twice.
• In the division column, under the number 5, write the number 2.
• Ask your child to multiply the number 5 by 2. The result of the multiplication is 10. This value must be written under the number 10. Next, you need to write a subtraction sign in the column. From 10 you need to subtract 10. You get 0.
• Write down in the column the number resulting from the subtraction - 0. 105 has a number left that was not involved in the division - 5. This number needs to be written down.
• The result is 5. This value must be divided by 5. The result is the number 1. This number must be written under 5. The result of the division is 21.

Parents need to explain that this division has no remainder.

You can start division with numbers 6,8,9, then go to 22, 44, 66 , and then to 232, 342, 345 , and so on.

Learning division with remainder

Once the child has mastered the material about division, you can make the task more difficult. Division with a remainder is the next step in learning. You need to explain using available examples:

• Invite your child to divide 35 by 8. Write the problem in a column.
• To make it as clear as possible for your child, you can show him the multiplication table. The table clearly shows that the number 35 includes the number 8 4 times.
• Write down the number 32 under the number 35.
• The child needs to subtract 32 from 35. The result is 3. The number 3 is the remainder.

Simple examples for a child

We can continue with the same example:

• When dividing 35 by 8, the remainder is 3. You need to add 0 to the remainder. In this case, after the number 4 in the column you need to put a comma. Now the result will be fractional.
• When dividing 30 by 8, the result is 3. This number must be written after the decimal point.
• Now you need to write 24 under the value 30 (the result of multiplying 8 by 3). The result will be 6. You also need to add a zero to the number 6. The result is 60.
• The number 60 contains the number 8 included 7 times. That is, it turns out to be 56.
• When subtracting 60 from 56, the result is 4. This number also needs to be signed 0. The result is 40. In the multiplication table, a child can see that 40 is the result of multiplying 8 by 5. That is, the number 40 includes the number 8 5 times. There is no remainder. The answer looks like this - 4.375.

This example may seem difficult to a child. Therefore, you need to divide values ​​that will have a remainder many times.

Teaching division using games

Parents can use division games to teach their students. You can give your child coloring books in which you need to determine the color of a pencil by dividing. You need to choose coloring pages with easy examples so that the child can solve the examples in his head.

The picture will be divided into parts containing the results of the division. And the colors to use will be examples. For example, the color red is labeled with an example: 15 divided by 3. You get 5. You need to find the part of the picture under this number and color it. Math coloring pages captivate children. Therefore, parents should try this method of teaching.

Learning to divide by column the smallest number by the largest

Division by this method assumes that the quotient will start at 0 and be followed by a comma.

In order for the student to correctly assimilate the information received, he needs to give an example of such a plan.

Division is one of the four basic mathematical operations (addition, subtraction, multiplication). Division, like other operations, is important not only in mathematics, but also in everyday life. For example, you as a whole class (25 people) donate money and buy a gift for the teacher, but you don’t spend it all, there will be change left over. So you will need to divide the change among everyone. The division operation comes into play to help you solve this problem.

Division is an interesting operation, as we will see in this article!

Dividing numbers

So, a little theory, and then practice! What is division? Division is breaking something into equal parts. That is, it could be a bag of sweets that needs to be divided into equal parts. For example, there are 9 candies in a bag, and the person who wants to receive them is three. Then you need to divide these 9 candies among three people.

It is written like this: 9:3, the answer will be the number 3. That is, dividing the number 9 by the number 3 shows the number of three numbers contained in the number 9. The reverse action, a check, will be multiplication. 3*3=9. Right? Absolutely.

So let's look at example 12:6. First, let's name each component of the example. 12 – dividend, that is. a number that can be divided into parts. 6 is a divisor, this is the number of parts into which the dividend is divided. And the result will be a number called “quotient”.

Let's divide 12 by 6, the answer will be the number 2. You can check the solution by multiplying: 2*6=12. It turns out that the number 6 is contained 2 times in the number 12.

Division with remainder

What is division with a remainder? This is the same division, only the result is not an even number, as shown above.

For example, let's divide 17 by 5. Since the largest number divisible by 5 to 17 is 15, then the answer will be 3 and the remainder is 2, and is written like this: 17:5 = 3(2).

For example, 22:7. In the same way, we determine the maximum number divisible by 7 to 22. This number is 21. The answer then will be: 3 and the remainder 1. And it is written: 22:7 = 3 (1).

Division by 3 and 9

A special case of division would be division by the number 3 and the number 9. If you want to find out whether a number is divisible by 3 or 9 without a remainder, then you will need:

Find the sum of the digits of the dividend.

Divide by 3 or 9 (depending on what you need).

If the answer is obtained without a remainder, then the number will be divided without a remainder.

For example, the number 18. The sum of the digits is 1+8 = 9. The sum of the digits is divisible by both 3 and 9. The number 18:9=2, 18:3=6. Divided without remainder.

For example, the number 63. The sum of the digits is 6+3 = 9. Divisible by both 9 and 3. 63:9 = 7, and 63:3 = 21. Such operations are carried out with any number to find out whether it is divisible with the remainder by 3 or 9, or not.

Multiplication and division

Multiplication and division are opposite operations. Multiplication can be used as a test for division, and division can be used as a test for multiplication. You can learn more about multiplication and master the operation in our article about multiplication. Which describes multiplication in detail and how to do it correctly. There you will also find the multiplication table and examples for training.

Here is an example of checking division and multiplication. Let's say the example is 6*4. Answer: 24. Then let's check the answer by division: 24:4=6, 24:6=4. It was decided correctly. In this case, the check is performed by dividing the answer by one of the factors.

Or an example is given for the division 56:8. Answer: 7. Then the test will be 8*7=56. Right? Yes. In this case, the test is performed by multiplying the answer by the divisor.

Division 3rd grade

In third grade they are just starting to go through division. Therefore, third graders solve the simplest problems:

Problem 1. A factory worker was given the task of putting 56 cakes into 8 packages. How many cakes should be put in each package to make the same amount in each?

Problem 2. On New Year's Eve at school, children in a class of 15 students were given 75 candies. How many candies should each child receive?

Problem 3. Roma, Sasha and Misha picked 27 apples from the apple tree. How many apples will each person get if they need to be divided equally?

Problem 4. Four friends bought 58 cookies. But then they realized that they could not divide them equally. How many additional cookies do the kids need to buy so that each gets 15?

Division 4th grade

The division in the fourth grade is more serious than in the third. All calculations are carried out using the column division method, and the numbers involved in the division are not small. What is long division? You can find the answer below:

Column division

What is long division? This is a method that allows you to find the answer to dividing large numbers. If prime numbers like 16 and 4 can be divided, and the answer is clear - 4. Then 512:8 is not easy for a child in his mind. And it’s our task to talk about the technique for solving such examples.

Let's look at an example, 512:8.

1 step. Let's write the dividend and divisor as follows:

The quotient will ultimately be written under the divisor, and the calculations under the dividend.

Step 2. We start dividing from left to right. First we take the number 5:

Step 3. The number 5 is less than the number 8, which means it will not be possible to divide. Therefore, we take another digit of the dividend:

Now 51 is greater than 8. This is an incomplete quotient.

Step 4. We put a dot under the divisor.

Step 5. After 51 there is another number 2, which means there will be one more number in the answer, that is. quotient is a two-digit number. Let's put the second point:

Step 6. We begin the division operation. The largest number divisible by 8 without a remainder to 51 is 48. Dividing 48 by 8, we get 6. Write the number 6 instead of the first dot under the divisor:

Step 7. Then write down the number exactly below the number 51 and put a “-” sign:

Step 8. Then we subtract 48 from 51 and get the answer 3.

* 9 step*. We take down the number 2 and write it next to the number 3:

Step 10 We divide the resulting number 32 by 8 and get the second digit of the answer – 4.

So the answer is 64, without remainder. If we divided the number 513, then the remainder would be one.

Division of three digits

Dividing three-digit numbers is done using the long division method, which was explained in the example above. An example of just a three-digit number.

Division of fractions

Dividing fractions is not as difficult as it seems at first glance. For example, (2/3):(1/4). The method of this division is quite simple. 2/3 is the dividend, 1/4 is the divisor. You can replace the division sign (:) with multiplication ( ), but to do this you need to swap the numerator and denominator of the divisor. That is, we get: (2/3)(4/1), (2/3)*4, this is equal to 8/3 or 2 integers and 2/3. Let's give another example, with an illustration for better understanding. Consider the fractions (4/7):(2/5):

As in the previous example, we reverse the 2/5 divisor and get 5/2, replacing division with multiplication. We then get (4/7)*(5/2). We make a reduction and answer: 10/7, then take out the whole part: 1 whole and 3/7.

Dividing numbers into classes

Let's imagine the number 148951784296, and divide it into three digits: 148,951,784,296. So, from right to left: 296 is the class of units, 784 is the class of thousands, 951 is the class of millions, 148 is the class of billions. In turn, in each class 3 digits have their own digit. From right to left: the first digit is units, the second digit is tens, the third is hundreds. For example, the class of units is 296, 6 is units, 9 is tens, 2 is hundreds.

Division of natural numbers

Division of natural numbers is the simplest division described in this article. It can be either with or without a remainder. The divisor and dividend can be any non-fractional, integer numbers.

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Division presentation

Presentation is another way to visualize the topic of division. Below we will find a link to an excellent presentation that does a good job of explaining how to divide, what division is, what dividend, divisor and quotient are. Don’t waste your time, but consolidate your knowledge!

Examples for division

Easy level

Average level

Difficult level

Games for developing mental arithmetic

Special educational games developed with the participation of Russian scientists from Skolkovo will help improve mental arithmetic skills in an interesting game form.

Game "Guess the operation"

The game “Guess the Operation” develops thinking and memory. The main point of the game is to choose a mathematical sign for the equality to be true. Examples are given on the screen, look carefully and put the required “+” or “-” sign so that the equality is true. The “+” and “-” signs are located at the bottom of the picture, select the desired sign and click on the desired button. If you answered correctly, you score points and continue playing.

Game "Simplification"

The game “Simplification” develops thinking and memory. The main essence of the game is to quickly perform a mathematical operation. A student is drawn on the screen at the blackboard, and a mathematical operation is given; the student needs to calculate this example and write the answer. Below are three answers, count and click the number you need using the mouse. If you answered correctly, you score points and continue playing.

Game "Quick addition"

The game "Quick Addition" develops thinking and memory. The main essence of the game is to choose numbers whose sum is equal to a given number. In this game, a matrix from one to sixteen is given. A given number is written above the matrix; you need to select the numbers in the matrix so that the sum of these digits is equal to the given number. If you answered correctly, you score points and continue playing.

Visual Geometry Game

The game "Visual Geometry" develops thinking and memory. The main essence of the game is to quickly count the number of shaded objects and select it from the list of answers. In this game, blue squares are shown on the screen for a few seconds, you need to quickly count them, then they close. Below the table there are four numbers written, you need to select one correct number and click on it with the mouse. If you answered correctly, you score points and continue playing.

Game "Piggy Bank"

The Piggy Bank game develops thinking and memory. The main essence of the game is to choose which piggy bank has more money. In this game there are four piggy banks, you need to count which piggy bank has the most money and show this piggy bank with the mouse. If you answered correctly, then you score points and continue playing.

Game "Fast addition reload"

The game “Fast addition reboot” develops thinking, memory and attention. The main point of the game is to choose the correct terms, the sum of which will be equal to the given number. In this game, three numbers are given on the screen and a task is given, add the number, the screen indicates which number needs to be added. You select the desired numbers from three numbers and press them. If you answered correctly, then you score points and continue playing.

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Teaching your child long division is easy. It is necessary to explain the algorithm of this action and consolidate the material covered.

• According to the school curriculum, division by columns begins to be explained to children in the third grade. Students who grasp everything on the fly quickly understand this topic
• But, if the child got sick and missed math lessons, or he did not understand the topic, then the parents must explain the material to the child themselves. It is necessary to convey information to him as clearly as possible
• Moms and dads must be patient during the child’s educational process, showing tact towards their child. Under no circumstances should you shout at a child if he doesn’t succeed in something, because this can discourage him from doing anything.

Important: In order for a child to understand the division of numbers, he must thoroughly know the multiplication table. If your child doesn't know multiplication well, he won't understand division.

During extracurricular activities at home, you can use cheat sheets, but the child must learn the multiplication table before starting the topic “Division.”

So, how to explain to a child division by column:

• Try to explain in small numbers first. Take counting sticks, for example 8 pieces
• Ask your child how many pairs are there in this row of sticks? Correct - 4. So, if you divide 8 by 2, you get 4, and when you divide 8 by 4, you get 2
• Let the child divide another number himself, for example, a more complex one: 24:4
• When the baby has mastered dividing prime numbers, then you can move on to dividing three-digit numbers into single-digit numbers.

Division is always a little more difficult for children than multiplication. But diligent additional studies at home will help the child understand the algorithm of this action and keep up with his peers at school.

Start with something simple—dividing by a single digit number:

Important: Calculate in your head so that the division comes out without a remainder, otherwise the child may get confused.

For example, 256 divided by 4:

• Draw a vertical line on a piece of paper and divide it in half from the right side. Write the first number on the left and the second number on the right above the line.
• Ask your child how many fours fit in a two - not at all
• Then we take 25. For clarity, separate this number from above with a corner. Again ask the child how many fours fit in twenty-five? That's right - six. We write the number “6” in the lower right corner under the line. The child must use the multiplication table to get the correct answer.
• Write down the number 24 under 25 and underline it to write down the answer - 1
• Ask again: how many fours can fit in a unit - not at all. Then we bring down the number “6” to one
• It turned out 16 - how many fours fit in this number? Correct - 4. Write “4” next to “6” in the answer
• Under 16 we write 16, underline it and it turns out “0”, which means we divided correctly and the answer turned out to be “64”

Written division by two digits

When the child has mastered division by a single digit number, you can move on. Written division by a two-digit number is a little more difficult, but if the child understands how this action is performed, then it will not be difficult for him to solve such examples.

Important: Again, start explaining with simple steps. The child will learn to select numbers correctly and it will be easy for him to divide complex numbers.

Do this simple action together: 184:23 - how to explain:

• Let's first divide 184 by 20, it turns out to be approximately 8. But we do not write the number 8 in the answer, since this is a test number
• Let's check if 8 is suitable or not. We multiply 8 by 23, we get 184 - this is exactly the number that is in our divisor. The answer will be 8

Important: For your child to understand, try taking 9 instead of 8, let him multiply 9 by 23, it turns out 207 - this is more than what we have in the divisor. The number 9 does not suit us.

So gradually the baby will understand division, and it will be easy for him to divide more complex numbers:

• Divide 768 by 24. Determine the first digit of the quotient - divide 76 not by 24, but by 20, we get 3. Write 3 in the answer under the line on the right
• Under 76 we write 72 and draw a line, write down the difference - it turns out 4. Is this number divisible by 24? No - we take down 8, it turns out 48
• Is 48 divisible by 24? That's right - yes. It turns out 2, write this number as the answer
• The result is 32. Now we can check whether we performed the division operation correctly. Do the multiplication in a column: 24x32, it turns out 768, then everything is correct

If the child has learned to divide by a two-digit number, then it is necessary to move on to the next topic. The algorithm for dividing by a three-digit number is the same as the algorithm for dividing by a two-digit number.

For example:

• Let's divide 146064 by 716. Take 146 first - ask your child whether this number is divisible by 716 or not. That's right - no, then we take 1460
• How many times can the number 716 fit in the number 1460? Correct - 2, so we write this number in the answer
• We multiply 2 by 716, we get 1432. We write this figure under 1460. The difference is 28, we write it under the line
• Let's take down 6. Ask your child - is 286 divisible by 716? That's right - no, so we write 0 in the answer next to 2. We also remove the number 4
• Divide 2864 by 716. Take 3 - a little, 5 - a lot, which means you get 4. Multiply 4 by 716, you get 2864
• Write 2864 under 2864, the difference is 0. Answer 204

Important: To check the correctness of division, multiply together with your child in a column - 204x716 = 146064. The division is done correctly.

The time has come to explain to the child that division can be not only whole, but also with a remainder. The remainder is always less than or equal to the divisor.

Division with a remainder should be explained using a simple example: 35:8=4 (remainder 3):

• How many eights fit in 35? Correct - 4. 3 left
• Is this number divisible by 8? That's right - no. It turns out the remainder is 3

After this, the child should learn that division can be continued by adding 0 to the number 3:

• The answer contains the number 4. After it we write a comma, since adding a zero indicates that the number will be a fraction
• It turns out 30. Divide 30 by 8, it turns out 3. Write it down, and under 30 we write 24, underline it and write 6
• We add the number 0 to number 6. Divide 60 by 8. Take 7 each, it turns out 56. Write under 60 and write down the difference 4
• To the number 4 we add 0 and divide by 8, we get 5 - write it down as the answer
• Subtract 40 from 40, we get 0. So, the answer is: 35:8 = 4.375

Advice: If your child doesn’t understand something, don’t get angry. Let a couple of days pass and try again to explain the material.

Mathematics lessons at school will also reinforce knowledge. Time will pass and the child will quickly and easily solve any division problems.

The algorithm for dividing numbers is as follows:

• Make an estimate of the number that will appear in the answer
• Find the first incomplete dividend
• Determine the number of digits in the quotient
• Find the numbers in each digit of the quotient
• Find the remainder (if there is one)

According to this algorithm, division is performed both by single-digit numbers and by any multi-digit number (two-digit, three-digit, four-digit, and so on).

When working with your child, often give him examples of how to perform the estimate. He must quickly calculate the answer in his head. For example:

• 1428:42
• 2924:68
• 30296:56
• 136576:64
• 16514:718

To consolidate the result, you can use the following division games:

• "Puzzle". Write five examples on a piece of paper. Only one of them must have the correct answer.

Condition for the child: Among several examples, only one was solved correctly. Find him in a minute.

Video: Arithmetic game for children addition, subtraction, division, multiplication

Video: Educational cartoon Mathematics Learning by heart the multiplication and division tables by 2