How to explain division by a single digit number to your child. How to divide into a column? How to explain long division to a child? Division by single-digit, two-digit, three-digit numbers, division with a remainder

Instructions

Before teaching how to divide two-digit numbers, you need to explain to your child that a number is the sum of tens and units. This will save him from future quite a common mistake that many children make. They begin to divide the first and second digits of the dividend and divisor by each other.

First, work from numbers to single digits. This technique is best practiced using knowledge of the multiplication table. The more such practice there is, the better. The skills of such division should be brought to automaticity, then it will be easier for the child to move on to the more complex topic of the two-digit divisor, which, like the dividend, is the sum of tens and units.

The most common method of dividing two-digit numbers is the brute method, which involves successively dividing numbers from 2 to 9 so that the resulting product equals the dividend. Example: divide 87 by 29. Reason as follows:

29 times 2 equals 54 – not enough;
29 x 3 = 87 – correct.

Draw the student's attention to the second digits (units) of the dividend and divisor, which are convenient to focus on when using the multiplication table. For example, in the above example, the second digit of the divisor is 9. Think about how much you need to multiply the number 9 so that the number of units of the product equals 7? Answer in in this case only one - by 3. This greatly simplifies the task of two-digit division. Test your guess by multiplying the entire number 29.

If the task is completed in writing, then it is advisable to use the column division method. This approach is similar to the previous one except that the student does not need to keep the numbers in his head and do mental calculations. It is better to arm yourself with a pencil or a rough piece of paper for written work.

Sources:

• multiplying two-digit numbers by two-digit tables

The topic of dividing numbers is one of the most important in the 5th grade math program. Without mastering this knowledge, further study of mathematics is impossible. Divide numbers happen in life every day. And you shouldn’t always rely on a calculator. To divide two numbers, you need to remember a certain sequence of actions.

You will need

• A sheet of paper in a square,
• pen or pencil

Instructions

Write down the dividend on one line. Separate them with a vertical line two lines high. Draw a horizontal line under the divisor and dividend perpendicular to the previous line. The quotient will be written to the right under this line. Below and to the left of the dividend, under the horizontal line, write down a zero.

Move the one leftmost, but not yet transferred, digit of the dividend down under the last horizontal line. Mark the transferred digit of the dividend with a dot.

Compare the number under the last horizontal line with the divisor. If the number is less than the divisor then continue from step 4, otherwise go to step 5.

Don't be upset if your child doesn't understand how to divide numbers in class. A teacher at school cannot always pay attention to every student. Be patient and become a home teacher for your student. Explain the mathematical process first in game form. Gradually move on to more complex tasks. The child will understand everything and mathematics will become his favorite subject.

Explaining division to a child in the form of a game

Put aside boring textbooks. Turn learning into an interesting game:

• take apples or candy. Ask your child to divide four candies or apples between two or three dolls or bears. Gradually increase the number of fruits to eight and ten. At first, the child will arrange objects slowly. Don't yell at him, be patient. If he’s wrong, calmly correct him. After the toys “receive” the candies, have the child count how many each doll has. Summarize. If there were 6 candies and they were distributed to three dolls, each got two. Explain that “sharing” means giving everyone an equal amount;
• another game example. We explain the division in numbers. Tell your child that the numbers are the same as apples or candies. Explain to him that the number of candies that needs to be divided is called the dividend. And the number of people into whom the candies are divided is the divisor;
• give the baby 6 apples. Ask him to distribute them equally to grandma, the cat and dad. Then let him divide the same number of items between the cat and grandmother. Explain why the results were different;
• Explain division with remainder. Give your baby 5 nuts and let him treat his dad and grandma to the same amount. The baby takes the remaining nut for himself. Using this example, explain that one nut is the remainder.

The above methods in a playful way will help the child understand the process of division and the fact that a larger number is divided by a smaller one. The first number is the number of apples or candies, and the second number is the participants between whom the items are divided. This information is enough for a child aged 5 to 8 years. Teach division to your child before school, it will be easier for him to learn math lessons in the future.

Explaining division to a child using the multiplication table as an example.

This teaching method is suitable for students primary classes, if they know multiplication. Tell us that division is the same as a multiplication table, but in it the opposite actions of multiplication occur. A good example for a child:

• multiply the number 5 by 4. You get 20;
• remind the student that the number 20 is the result of multiplying the two above numbers;
• divide 20 by 5. Get 4. This will clearly show that division is the opposite of multiplication.

Consider examples with other numbers. If a student has mastered the multiplication table well and understands the connection between two mathematical operations, division will be easy to master.

Explaining division to the child - definition of concepts

Explain to your child the names of the numbers involved in division:

• dividend. The number to be divided;
• divider. The number by which the dividend is divided;
• private. The result obtained after division.

For clarity, use the same examples with candy and people or toys that the child should treat with sweets.

Explaining division to a child

Proceed to this training only after the child has mastered the above methods. He should also know how numbers are multiplied into a column. Let's take a simple example: 110 divided by 5. Explanation process:

• write these numbers on a blank piece of paper;
• divide them with perpendicular lines as you would divide in a column;
• explain which number is a divisor and which is a dividend;
• Determine with your child which number can be used for division first. The first digit – 1 cannot be divided by 5. This means that you need to take the next digit to it and you get the number 11. The number 5 can fit into 11 twice;
• write the number 2 in the column under the five. Ask the child to multiply 5 by 2. The result is 10. Write this figure under the number 11;
• Together with your child, subtract the number 10 from 11. You get 1. Write the remaining zero in the column next to the one. It turns out 10;
• Divide 10 by 5 with your child. You get 2. Write this number under the five, and the final result is 22.

Start learning with two-digit or even single-digit numbers that can be divided without a remainder. Gradually make the task more difficult.

To make it easier for your child to learn mathematics, arouse his interest in this lesson. Now division tables have appeared. But does a child need to memorize it if he knows the multiplication table and understands that division is a process in reverse? Everything depends not only on school teacher, but also from your activities with the student.

Column? How can you independently practice the skill of long division at home if your child did not learn something at school? Dividing by columns is taught in grades 2-3; for parents, of course, this is a passed stage, but if you wish, you can remember the correct notation and explain in an understandable way to your student what he will need in life.

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What should a 2nd-3rd grade child know to learn to do long division?

How to correctly explain division to a 2-3 grade child so that he doesn’t have problems in the future? First, let's check if there are any gaps in knowledge. Make sure that:

• the child can freely perform addition and subtraction operations;
• knows the digits of numbers;
• knows by heart.

How to explain to a child the meaning of the action “division”?

• Everything needs to be explained to the child using a clear example.

Ask to share something among family members or friends. For example, candy, pieces of cake, etc. It is important that the child understands the essence - you need to divide equally, i.e. without a trace. Practice with different examples.

Let's say 2 groups of athletes must take seats on the bus. We know how many athletes are in each group and how many seats there are on the bus. You need to find out how many tickets one and the other group need to buy. Or 24 notebooks should be distributed to 12 students, as many as each gets.

• When the child understands the essence of the principle of division, show the mathematical notation of this operation and name the components.
• Explain that Division is the opposite operation of multiplication, multiplication inside out.

It is convenient to show the relationship between division and multiplication using a table as an example.

For example, 3 times 4 equals 12.
3 is the first multiplier;
4 - second factor;
12 is the product (the result of multiplication).

If 12 (the product) is divided by 3 (the first factor), we get 4 (the second factor).

Components when divided are called differently:

12 - dividend;
3 - divider;
4 - quotient (result of division).

How to explain to a child the division of a two-digit number by a single-digit number not in a column?

For us adults, it’s easier to write “in the corner” the old fashioned way – and that’s the end of it. BUT! Children have not yet completed long division, what should they do? How to teach a child to divide two-digit number to unambiguous without using column notation?

Let's take 72:3 as an example.

It's simple! We break down 72 into numbers that can easily be divided verbally by 3:
72=30+30+12.

Everything immediately became clear: we can divide 30 by 3, and a child can easily divide 12 by 3.
All that remains is to add up the results, i.e. 72:3=10 (obtained when 30 was divided by 3) + 10 (30 divided by 3) + 4 (12 divided by 3).

72:3=24
We did not use long division, but the child understood the reasoning and completed the calculations without difficulty.

After simple examples, you can move on to studying long division and teach your child to correctly write examples in a “corner”. To begin with, use only examples of division without a remainder.

How to explain long division to a child: solution algorithm

Large numbers are difficult to divide in your head; it is easier to use column division notation. To teach your child to perform calculations correctly, follow the algorithm:

• Determine where the dividend and divisor are in the example. Ask your child to name the numbers (what we will divide by what).

213:3
213 - dividend
3 - divider

• Write down the dividend - "corner" - divisor.

• Determine which part of the dividend we can use to divide by a given number.

We reason like this: 2 is not divisible by 3, which means we take 21.

• Determine how many times the divisor “fits” in the selected part.

21 divided by 3 - take 7.

• Multiply the divisor by the selected number, write the result under the “corner”.

7 multiplied by 3 - we get 21. Write it down.

• Find the difference (remainder).

At this stage of reasoning, teach your child to check himself. It is important that he understands that the result of a subtraction must ALWAYS be less than the divisor. If it doesn't work out, you need to increase the selected number and perform the action again.

• Repeat the steps until the remainder is 0.

How to reason correctly to teach a 2-3 grade child to divide by column

How to explain division to a child 204:12=?
1. Write it down in a column.
204 is the dividend, 12 is the divisor.

2. 2 is not divisible by 12, so we take 20.
3. To divide 20 by 12, take 1. Write 1 under the “corner”.
4. 1 multiplied by 12 gets 12. We write it under 20.
5. 20 minus 12 gets 8.
Let's check ourselves. Is 8 less than 12 (divisor)? Ok, that's right, let's move on.

6. Next to 8 we write 4. 84 divided by 12. How much should we multiply 12 to get 84?
It’s hard to say right away, we’ll try to use the selection method.
Let's take 8, for example, but don't write them down yet. We count verbally: 8 multiplied by 12 equals 96. And we have 84! Doesn't fit.
Let's try smaller ones... For example, let's take 6. We check ourselves verbally: 6 multiplied by 12 equals 72. 84-72=12. We got the same number as our divisor, but it should be either zero or less than 12. So the optimal number is 7!

7. We write 7 under the “corner” and perform the calculations. 7 multiplied by 12 gives 84.
8. We write the result in a column: 84 minus 84 equals zero. Hooray! We decided correctly!

So, you have taught your child to divide by column, now all that remains is to practice this skill and bring it to automatism.

Why is it difficult for children to learn long division?

Remember that problems with mathematics arise from the inability to quickly do simple arithmetic operations. IN primary school you need to practice and make addition and subtraction automatic, and learn the multiplication table from cover to cover. All! The rest is a matter of technique, and it is developed with practice.

Be patient, do not be lazy, once again explain to the child what he did not learn in the lesson, tediously but meticulously understand the reasoning algorithm and talk through each intermediate operation before voicing a ready answer. Give additional examples to practice skills, play math games- this will bear fruit and you will see the results and rejoice at your child’s success very soon. Be sure to show where and how you can apply the acquired knowledge in everyday life.

Dear readers! Tell us how you teach your children to do long division, what difficulties you have encountered and how you have overcome them.

Unfortunately, modern educational program does not always involve explaining every topic to students, especially something as complex as long division. In such cases, parents themselves have to teach students at home.

Step-by-step instructions for learning long division

First, you need to determine the child’s basis: repeat with him the names of the division elements (dividend, divisor, quotient, remainder), the digits of the number and the multiplication table. Without this knowledge, the child will not be able to master division. First you need to show the operation on simple examples from the multiplication table, that is, 56: 7 = 8. Next, show an example of division three-digit number without a remainder, when the first digit of the dividend is greater than the divisor, for example, 422: 2. It is necessary to divide each digit in order by the divisor as follows: 4 divided by 2 will be 2, write, 2 by 2 is 1, write, 2 by 2 - again one, let's write it down. The result was 211. The result must be double-checked by reverse multiplication.

Learning long division requires practice and repetition of each step. Pick up a few more of the same simple operations, for example, 936 divided by 3, 488 divided by 4, etc. Comment on your actions the same way each time, so that they are imprinted in the child’s head, and he repeats them to himself while dividing:

• Take the first digit of the number and divide it by the divisor. How many times can a divisor appear in the dividend?
• If the first digit is less than the divisor, take the number from the first two digits, divide, and write down the result.
• We multiply the divisor by the quotient and subtract it from the dividend, sign the result of the subtraction.
• We take down the next digit of the dividend: can it be divided by the divisor? If not, then we take down another number and divide, writing down the result.
• We multiply the last digit of the quotient by the divisor and subtract it from the remaining dividend. We get the rest.

Using an example, it looks like this: divide 563 by 11. 5 cannot be divided by 11, take 56. 11 can fit 5 times into 56, write it as a quotient. 5 multiplied by 11 equals 55. 56 minus 55 equals 1. 1 cannot be divided by 11, so take away 3. 13 will only fit 11 once, write it down. 1 multiplied by 11 is 11, subtracted from 13, we get 2. Answer: quotient 51, remainder 2.

It is very important that the child correctly signs the result of the subtraction and takes down the numbers, and each digit of the quotient is always determined only by selecting the numbers. Work with your child regularly, but not for very long: gradually he will get better at cracking problems like nuts.

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 with characteristic examples of division by a column of natural numbers with detailed explanations progress of the solution and illustrations.

Page navigation.

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 - like this less chance get lost in the correct 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 entry when dividing into a column it will be like this:

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 the natural number 614,808 by 51,234 with a column (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 digits 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 record will take the following form:

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.

On 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 a number is obtained 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 (rest. 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:

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