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 yell at your 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. Ask the child again 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

Read the topic of the lesson: “Division with a remainder.” What do you already know about this topic?

Can you distribute 8 plums equally on two plates (Fig. 1)?

Rice. 1. Illustration for example

You can put 4 plums in each plate (Fig. 2).

Rice. 2. Illustration for example

The action we performed can be written like this.

8: 2 = 4

Do you think it is possible to divide 8 plums equally onto 3 plates (Fig. 3)?

Rice. 3. Illustration for example

Let's act like this. First, put one plum in each plate, then a second plum. We will have 2 plums left, but 3 plates. This means that we cannot distribute them evenly further. We put 2 plums in each plate, and we had 2 plums left (Fig. 4).

Rice. 4. Illustration for example

Let's continue observing.

Read the numbers. Among the given numbers, find those that are divisible by 3.

11, 12, 13, 14, 15, 16, 17, 18, 19

Test yourself.

The remaining numbers (11, 13, 14, 16, 17, 19) are not divisible by 3, or they say "shared with the remainder."

Let's find the value of the quotient.

Let's find out how many times 3 is contained in the number 17 (Fig. 5).

Rice. 5. Illustration for example

We see that 3 ovals fit 5 times and 2 ovals remain.

The completed action can be written like this.

17: 3 = 5 (remaining 2)

You can also write it in a column (Fig. 6)

Rice. 6. Illustration for example

Look at the pictures. Explain the captions to these figures (Fig. 7).

Rice. 7. Illustration for example

Let's look at the first picture (Fig. 8).

Rice. 8. Illustration for example

We see that 15 ovals were divided into 2. 2 were repeated 7 times, with the remainder being 1 oval.

Let's look at the second picture (Fig. 9).

Rice. 9. Illustration for example

In this figure, 15 squares were divided into 4. 4 were repeated 3 times, with the remainder being 3 squares.

Let's look at the third picture (Fig. 10).

Rice. 10. Illustration for example

We can say that 15 ovals were divided into 3. 3 were repeated 5 times equally. In such cases the remainder is said to be 0.

Let's do the division.

We divide seven squares into three. We get two groups, and one square remains. Let's write down the solution (Fig. 11).

Rice. 11. Illustration for example

Let's do the division.

Let's find out how many times four are contained in the number 10. We see that the number 10 contains four times 2 times and 2 squares remain. Let's write down the solution (Fig. 12).

Rice. 12. Illustration for example

Let's do the division.

Let's find out how many times two are contained in the number 11. We see that in the number 11 two are contained 5 times and 1 square remains. Let's write down the solution (Fig. 13).

Rice. 13. Illustration for example

Let's draw a conclusion. Dividing with a remainder means finding out how many times the divisor is contained in the dividend and how many units are left.

Division with a remainder can also be performed on the number line.

On the number line we mark segments of 3 divisions and see that there are three divisions three times and one division remains (Fig. 14).

Rice. 14. Illustration for example

Let's write down the solution.

10: 3 = 3 (remaining 1)

Let's do the division.

On the number line we mark segments of 3 divisions and see that there are three divisions three times and two divisions remain (Fig. 15).

Rice. 15. Illustration for example

Let's write down the solution.

11: 3 = 3 (remaining 2)

Let's do the division.

On the number line we mark segments of 3 divisions and see that we got exactly 4 times, there is no remainder (Fig. 16).

Rice. 16. Illustration for example

Let's write down the solution.

12: 3 = 4

Today in the lesson we got acquainted with division with a remainder, learned how to perform the named action using a drawing and a number line, and practiced solving examples on the topic of the lesson.

Bibliography

1. M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 1. - M.: “Enlightenment”, 2012.
2. M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 2. - M.: “Enlightenment”, 2012.
3. M.I. Moro. Mathematics lessons: Methodological recommendations for teachers. 3rd grade. - M.: Education, 2012.
4. Regulatory document. Monitoring and evaluation of learning outcomes. - M.: “Enlightenment”, 2011.
5. “School of Russia”: Programs for primary school. - M.: “Enlightenment”, 2011.
6. S.I. Volkova. Mathematics: Test work. 3rd grade. - M.: Education, 2012.
7. V.N. Rudnitskaya. Tests. - M.: “Exam”, 2012.

1. Nsportal.ru ().
2. Prosv.ru ().
3. Do.gendocs.ru ().

Homework

1. Write down the numbers that are divisible by 2 without a remainder.

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19

2. Perform division with a remainder using a picture.

3. Perform division with a remainder using the number line.

4. Create an assignment for your friends on the topic of the lesson.

Column division(you can also find the name division corner) is a standard procedure inarithmetic, designed to divide simple or complex multi-digit numbers by breakingdivided into a number of simpler steps. As with all division problems, one number, calleddivisible, is divided into another, calleddivider, producing a result calledprivate.

The column can be used to divide natural numbers without a remainder, as well as to divide natural numbers with the remainder.

Rules for writing when dividing by a column.

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

First, the dividend and the divisor are written in one line from left to right, after which between the writtennumbers represent a symbol of the form.

For example, if the dividend is 6105 and the divisor is 55, then their correct notation when dividing inthe column will be like this:

Look at the following diagram illustrating places to write dividend, divisor, quotient,remainder and intermediate calculations when dividing by a column:

From the above diagram it is clear that the required quotient (or incomplete quotient when divided with a remainder) will bewritten below the divisor under the horizontal bar. And intermediate calculations will be carried out belowdivisible, and you need to take care in advance about the availability of space on the page. In this case, one should be guidedrule: the greater the difference in the number of characters in the entries of the dividend and the divisor, the greaterspace will be required.

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

How to do long division is best explained with an example.Calculate:

512:8=?

First, let's write down the dividend and divisor in a column. It will look like this:

We will write their quotient (result) under the divisor. For us this is number 8.

1. Define an incomplete quotient. 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 workwith this number. If this number is less than the divisor, then we need to add the following to considerationon the left the figure in the notation of the dividend, and work further with the number determined by the two consideredin numbers. For convenience, we highlight in our notation the number with which we will work.

2. Take 5. The number 5 is less than 8, which means you need to take one more number from the dividend. 51 is greater than 8. So.this is an incomplete quotient. We put a dot in the quotient (under the corner of the divisor).

After 51 there is only one number 2. This means we add one more point to the result.

3. Now, remembering multiplication table by 8, find the product closest to 51 → 6 x 8 = 48→ write the number 6 into the quotient:

We write 48 under 51 (if we multiply 6 from the quotient by 8 from the divisor, we get 48).

Attention! When writing under an incomplete quotient, the rightmost digit of the incomplete quotient should be aboverightmost digit works.

4. Between 51 and 48 on the left we put “-” (minus). Subtract according to the rules of subtraction in column 48 and below the lineLet's write down the result.

However, if the result of the subtraction is zero, then it does not need to be written (unless the subtraction is inthis point is not the very last action that completely completes the division process column).

The remainder is 3. Let's compare the remainder with the divisor. 3 is less than 8.

Attention!If the remainder is greater than the divisor, then we made a mistake in the calculation and the product iscloser than the one we took.

5. Now, under the horizontal line to the right of the numbers located there (or to the right of the place where we do notbegan to write down zero) we write down the number located in the same column in the record of the dividend. If inThere are no numbers in the dividend entry in this column, then division by column ends here.

The number 32 is greater than 8. And again, using the multiplication table by 8, we find the nearest product → 8 x 4 = 32:

The remainder was zero. This means that the numbers are completely divided (without remainder). If after the lastsubtraction results in zero, and there are no more digits left, then this is the remainder. We add it to the quotient inparentheses (eg 64(2)).

Column division of multi-digit natural numbers.

Division by a multi-digit natural number is done in a similar way. At the same time, in the firstThe “intermediate” dividend includes so many high-order digits that it becomes larger than the divisor.

For example, 1976 divided by 26.

• The number 1 in the most significant digit is less than 26, so consider a number made up of two digits senior ranks - 19.
• The number 19 is also less than 26, so consider a number made up of the digits of the three highest digits - 197.
• The number 197 is greater than 26, divide 197 tens by 26: 197: 26 = 7 (15 tens left).
• Convert 15 tens to units, add 6 units from the units digit, we get 156.
• Divide 156 by 26 to get 6.

So 1976: 26 = 76.

If at some division step the “intermediate” dividend turns out to be less than the divisor, then in the quotient0 is written, and the number from this digit is transferred to the next, lower digit.

Division with decimal fraction in quotient.

Decimals online. Converting decimals to fractions and fractions to decimals.

If the natural number is not divisible by a single digit natural number, you can continuebitwise division and get a decimal fraction in the quotient.

For example, divide 64 by 5.

• Divide 6 tens by 5, we get 1 ten and 1 ten as a remainder.
• We convert the remaining ten into units, add 4 from the ones category, and get 14.
• We divide 14 units by 5, we get 2 units and a remainder of 4 units.
• We convert 4 units to tenths, we get 40 tenths.
• Divide 40 tenths by 5 to get 8 tenths.

So 64:5 = 12.8

Thus, if, when dividing a natural number by a natural single-digit or multi-digit numberthe remainder is obtained, then you can put a comma in the quotient, convert the remainder into units of the following,smaller digit and continue dividing.

The easiest way to divide multi-digit numbers is with a column. Column division is also called corner division.

Before we begin to perform division by a column, we will consider in detail the very form of recording division by a column. First, write down the dividend and put a vertical line to the right of it:

Behind the vertical line, opposite the dividend, write the divisor and draw a horizontal line under it:

Under the horizontal line, the resulting quotient will be written step by step:

Intermediate calculations will be written under the dividend:

The full form of writing division by column is as follows:

How to divide by column

Let's say we need to divide 780 by 12, write the action in a column and proceed to division:

Column division is performed in stages. The first thing we need to do is determine the incomplete dividend. We look at the first digit of the dividend:

this number is 7, since it is less than the divisor, we cannot start division from it, which means we need to take another digit from the dividend, the number 78 is greater than the divisor, so we start division from it:

In our case the number 78 will be incomplete divisible, it is called incomplete because it is only a part of the divisible.

Having determined the incomplete dividend, we can find out how many digits will be in the quotient, for this we need to calculate how many digits are left in the dividend after the incomplete dividend, in our case there is only one digit - 0, this means that the quotient will consist of 2 digits.

Having found out the number of digits that should be in the quotient, you can put dots in its place. If, when completing the division, the number of digits turns out to be more or less than the indicated points, then an error was made somewhere:

Let's start dividing. We need to determine how many times 12 is contained in the number 78. To do this, we sequentially multiply the divisor by the natural numbers 1, 2, 3, ... until we get a number as close as possible to the incomplete dividend or equal to it, but not exceeding it. Thus, we get the number 6, write it under the divisor, and from 78 (according to the rules of column subtraction) we subtract 72 (12 6 = 72). After we subtract 72 from 78, the remainder is 6:

Please note that the remainder of the division shows us whether we have chosen the number correctly. If the remainder is equal to or greater than the divisor, then we did not choose the number correctly and we need to take a larger number.

To the resulting remainder - 6, add the next digit of the dividend - 0. As a result, we get an incomplete dividend - 60. Determine how many times 12 is contained in the number 60. We get the number 5, write it in the quotient after the number 6, and subtract 60 from 60 ( 12 5 = 60). The remainder is zero:

Since there are no more digits left in the dividend, it means 780 is divided by 12 completely. As a result of performing long division, we found the quotient - it is written under the divisor:

Let's consider an example when the quotient turns out to be zeros. Let's say we need to divide 9027 by 9.

We determine the incomplete dividend - this is the number 9. We write 1 into the quotient and subtract 9 from 9. The remainder is zero. Usually, if in intermediate calculations the remainder is zero, it is not written down:

We take down the next digit of the dividend - 0. We remember that when dividing zero by any number there will be zero. We write zero into the quotient (0: 9 = 0) and subtract 0 from 0 in intermediate calculations. Usually, in order not to clutter up intermediate calculations, calculations with zero are not written:

We take down the next digit of the dividend - 2. In intermediate calculations it turned out that the incomplete dividend (2) is less than the divisor (9). In this case, write zero to the quotient and remove the next digit of the dividend:

We determine how many times 9 is contained in the number 27. We get the number 3, write it as a quotient, and subtract 27 from 27. The remainder is zero:

Since there are no more digits left in the dividend, it means that the number 9027 is divided by 9 completely:

Let's consider an example when the dividend ends in zeros. Let's say we need to divide 3000 by 6.

We determine the incomplete dividend - this is the number 30. We write 5 into the quotient and subtract 30 from 30. The remainder is zero. As already mentioned, it is not necessary to write zero in the remainder in intermediate calculations:

We take down the next digit of the dividend - 0. Since dividing zero by any number will result in zero, we write zero in the quotient and subtract 0 from 0 in intermediate calculations:

We take down the next digit of the dividend - 0. We write another zero into the quotient and subtract 0 from 0 in intermediate calculations. Since in intermediate calculations the calculation with zero is usually not written down, the entry can be shortened, leaving only the remainder - 0. Zero in the remainder in at the very end of the calculation is usually written to show that the division is complete:

Since there are no more digits left in the dividend, it means 3000 is divided by 6 completely:

Column division with remainder

Let's say we need to divide 1340 by 23.

We determine the incomplete dividend - this is the number 134. We write 5 into the quotient and subtract 115 from 134. The remainder is 19:

We take down the next digit of the dividend - 0. We determine how many times 23 is contained in the number 190. We get the number 8, write it into the quotient, and subtract 184 from 190. We get the remainder 6:

Since there are no more digits left in the dividend, the division is over. The result is an incomplete quotient of 58 and a remainder of 6:

1340: 23 = 58 (remainder 6)

It remains to consider an example of division with a remainder, when the dividend is less than the divisor. Let us need to divide 3 by 10. We see that 10 is never contained in the number 3, so we write 0 as a quotient and subtract 0 from 3 (10 · 0 = 0). Draw a horizontal line and write down the remainder - 3:

3: 10 = 0 (remainder 3)

Long division calculator

This calculator will help you perform long division. Simply enter the dividend and divisor and click the Calculate button.

We will move on from the general idea of ​​dividing natural numbers with a remainder, and in this article we will understand the principles by which this action is carried out. At all division with remainder has a lot in common with dividing natural numbers without a remainder, so we will often refer to the material in this article.

First, let's look at dividing natural numbers with a remainder. Next we will show how you can find the result of dividing natural numbers with a remainder by performing sequential subtraction. After this, we will move on to the method of selecting an incomplete quotient, not forgetting to give examples with a detailed description of the solution. Next, we will write an algorithm that allows us to divide natural numbers with a remainder in the general case. At the end of the article, we will show how to check the result of dividing natural numbers with a remainder.

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Dividing natural numbers with a remainder

One of the most convenient ways to divide natural numbers with a remainder is long division. In the article Dividing Natural Numbers by Columns, we discussed this division method in great detail. We will not repeat ourselves here, but simply give the solution to one example.

Example.

Divide with the remainder of the natural number 273,844 by the natural number 97.

Solution.

Let's do the division by column:

Thus, the partial quotient of 273,844 divided by 97 is 2,823, and the remainder is 13.

Answer:

273,844:97=2,823 (rest. 13) .

Dividing natural numbers with a remainder through sequential subtraction

You can find the partial quotient and remainder when dividing natural numbers by sequentially subtracting the divisor.

The essence of this approach is simple: sets with the required number of elements are sequentially formed from the elements of the existing set until this is possible, the number of resulting sets gives the incomplete quotient, and the number of remaining elements in the original set is the remainder of the division.

Let's give an example.

Example.

Let's say we need to divide 7 by 3.

Solution.

Let's imagine that we need to put 7 apples into bags of 3 apples. From the original number of apples, we take 3 pieces and put them in the first bag. In this case, due to the meaning of subtracting natural numbers, we are left with 7−3=4 apples. We again take 3 of them and put them in the second bag. After this we are left with 4−3=1 apple. It is clear that this is where the process ends (we cannot form another package with the required number of apples, since the remaining number of apples 1 is less than the quantity 3 we need). As a result, we have two bags with the required number of apples and one apple left.

Then, due to the meaning of dividing natural numbers with a remainder, we can say that we got the following result 7:3=2 (rest. 1).

Answer:

7:3=2 (rest. 1) .

Let's consider the solution to another example, and we will only give mathematical calculations.

Example.

Divide the natural number 145 by 46 using sequential subtraction.

Solution.

145−46=99 (if necessary, refer to the article subtraction of natural numbers). Since 99 is greater than 46, we subtract the divisor a second time: 99−46=53. Since 53>46, we subtract the divisor a third time: 53−46=7. Since 7 is less than 46, we will not be able to carry out the subtraction again, that is, this ends the process of sequential subtraction.

As a result, we needed to successively subtract the divisor 46 from the dividend 145 3 times, after which we got the remainder 7. Thus, 145:46=3 (remaining 7).

Answer:

145:46=3 (remaining 7) .

It should be noted that if the dividend is less than the divisor, then we will not be able to carry out sequential subtraction. Yes, this is not necessary, since in this case we can immediately write the answer. In this case, the partial quotient is equal to zero, and the remainder is equal to the dividend. That is, if a

It must also be said that dividing natural numbers with a remainder using the method considered is good only when a small number of successive subtractions are required to obtain the result.

Selection of incomplete quotient

When dividing given natural numbers a and b with a remainder, the partial quotient c can be found. Now we will show what the selection process is based on and how it should proceed.

First, let's decide among which numbers to look for the incomplete quotient. When we talked about the meaning of dividing natural numbers with a remainder, we found out that an incomplete quotient can be either zero or a natural number, that is, one of the numbers 0, 1, 2, 3, ... Thus, the required incomplete quotient is one of the written numbers, and we just have to go through them to determine which number the partial quotient is.

Next, we will need an equation of the form d=a−b·c, which specifies , as well as the fact that the remainder is always less than the divisor (we also mentioned this when we talked about the meaning of dividing natural numbers with a remainder).

Now we can proceed directly to the description of the process of selecting an incomplete quotient. The dividend a and the divisor b are known to us initially; as an incomplete quotient c we successively take the numbers 0, 1, 2, 3, ..., each time calculating the value d=a−b·c and comparing it with the divisor. This process ends as soon as the resulting value is less than the divisor. In this case, the number c at this step is the desired incomplete quotient, and the value d=a−b·c is the remainder of the division.

It remains to analyze the process of selecting an incomplete quotient using an example.

Example.

Divide with the remainder of the natural number 267 by 21.

Solution.

Let's select an incomplete quotient. In our example, a=267, b=21. We will successively assign c the values ​​0, 1, 2, 3, ..., calculating at each step the value d=a−b·c and comparing it with the divisor 21.

At c=0 we have d=a−b·c=267−21·0=267−0=267(first multiplication of natural numbers is performed, and then subtraction, this is written in the article). The resulting number is greater than 21 (if necessary, study the material in the article comparing natural numbers). Therefore, we continue the selection process.

At c=1 we have d=a−b·c=267−21·1=267−21=246. Since 246>21, we continue the process.

At c=2 we get d=a−b·c=267−21·2=267−42=225. Since 225>21, we move on.

At c=3 we have d=a−b·c=267−21·3=267−63=204. Since 204>21, we continue the selection.

At c=12 we get d=a−b·c=267−21·12=267−252=15. We received the number 15, which is less than 21, so the process can be considered complete. We selected the incomplete quotient c=12, with the remainder d equal to 15.

Answer:

267:21=12 (rest. 15) .

Algorithm for dividing natural numbers with a remainder, examples, solutions

In this section, we will consider an algorithm that allows division with a remainder of a natural number a by a natural number b in cases where the method of sequential subtraction (and the method of selecting an incomplete quotient) requires too many computational operations.

Let us immediately note that if the dividend a is less than the divisor b, then we know both the partial quotient and the remainder: for a b.

Before we describe in detail all the steps of the algorithm for dividing natural numbers with a remainder, we will answer three questions: what do we initially know, what do we need to find, and based on what considerations will we do this? Initially, we know the dividend a and the divisor b. We need to find the partial quotient c and the remainder d. The equality a=b·c+d defines the relationship between the dividend, divisor, partial quotient and remainder. From the written equality it follows that if we present the dividend a as a sum b·c+d, in which d is less than b (since the remainder is always less than the divisor), then we will see both the incomplete quotient c and the remainder d.

All that remains is to figure out how to represent the dividend a as a sum b·c+d. The algorithm for doing this is very similar to the algorithm for dividing natural numbers without a remainder. We will describe all the steps, and at the same time we will solve the example for greater clarity. Divide 899 by 47.

The first five points of the algorithm will allow you to represent the dividend as the sum of several terms. It should be noted that the actions from these points are repeated cyclically again and again until all the terms that add up to the dividend are found. In the final sixth point, the resulting sum is converted to the form b·c+d (if the resulting sum no longer has this form), from where the required incomplete quotient and remainder become visible.

So, let's start representing the dividend 899 as the sum of several terms.

First, we calculate how much more the number of digits in the dividend is greater than the number of digits in the divisor, and remember this number.

In our example, the dividend has 3 digits (899 is a three-digit number), and the divisor has two digits (47 is a two-digit number), therefore, the dividend has one more digit, and we remember the number 1.

Now in the divisor entry on the right we add the numbers 0 in the amount determined by the number obtained in the previous paragraph. Moreover, if the written number is greater than the dividend, then you need to subtract 1 from the number remembered in the previous paragraph.

Let's return to our example. In the notation of the divisor 47, we add one digit 0 to the right, and we get the number 470. Since 470<899 , то запомненное в предыдущем пункте число НЕ нужно уменьшать на 1 . Таким образом, у нас в памяти остается число 1 .

After this, to the number 1 on the right we assign the numbers 0 in an amount determined by the number memorized in the previous paragraph. In this case, we get a unit of digit, which we will work with further.

In our example, we assign 1 digit 0 to the number 1, and we get the number 10, that is, we will work with the tens place.

Now we successively multiply the divisor by 1, 2, 3, ... units of the working digit until we get a number greater than or equal to the dividend.

We found out that in our example the working digit is the tens digit. Therefore, we first multiply the divisor by one unit in the tens place, that is, multiply 47 by 10, we get 47 10 = 470. The resulting number 470 is less than the dividend 899, so we proceed to multiplying the divisor by two units in the tens place, that is, we multiply 47 by 20. We have 47·20=940. We got a number that is greater than 899.

The number obtained at the penultimate step during sequential multiplication is the first of the required terms.

In the example being analyzed, the required term is the number 470 (this number is equal to the product 47·100, we will use this equality later).

After this, we find the difference between the dividend and the first term found. If the resulting number is greater than the divisor, then we proceed to find the second term. To do this, we repeat all the described steps of the algorithm, but now we take the number obtained here as the dividend. If at this point we again obtain a number greater than the divisor, then we proceed to find the third term, once again repeating the steps of the algorithm, taking the resulting number as the dividend. And so we proceed further, finding the fourth, fifth and subsequent terms until the number obtained at this point is less than the divisor. As soon as this happens, we take the number obtained here as the last term we are looking for (looking ahead, let’s say that it is equal to the remainder), and move on to the final stage.

Let's return to our example. At this step we have 899−470=429. Since 429>47, we take this number as the dividend and repeat all stages of the algorithm with it.

The number 429 has one more digit than the number 47, so remember the number 1.

Now in the notation of the dividend on the right we add one digit 0, we get the number 470, which is greater than the number 429. Therefore, from the number 1 remembered in the previous paragraph, we subtract 1, we get the number 0, which we remember.

Since in the previous paragraph we remembered the number 0, then to the number 1 there is no need to assign a single digit 0 to the right. In this case, we have the number 1, that is, the working digit is the ones digit.

Now we sequentially multiply the divisor 47 by 1, 2, 3, ... We will not dwell on this in detail. Let's just say that 47·9=423<429 , а 47·10=470>429. The second term we are looking for is the number 423 (which is equal to 47 9, which we will use further).

The difference between 429 and 423 is 6. This number is less than the divisor 47, so it is the third (and last) term we are looking for. Now we can move on to the final stage.

Well, we have come to the final stage. All previous actions were aimed at presenting the dividend as the sum of several terms. Now the resulting sum remains to be converted to the form b·c+d. The distributive property of multiplication relative to addition will help us cope with this task. After this, the required incomplete quotient and remainder will become visible.

In our example, the dividend 899 is equal to the sum of three terms 470, 423 and 6. The sum 470+423+6 can be rewritten as 47·10+47·9+6 (remember, we paid attention to the equalities 470=47·10 and 423=47·9). Now we apply the property of multiplying a natural number by a sum, and we get 47·10+47·9+6= 47·(10+9)+6= 47·19+6. Thus, the dividend is transformed to the form we need 899=47·19+6, from which the incomplete quotient 19 and the remainder 6 can be easily found.

So, 899:47=19 (rest. 6).

Of course, when solving examples, you will not describe in such detail the process of division with a remainder.