>> Lesson 13. Division by two digits and three digit number

Divide 876 by 24. Calculating 800: 20 = 40 shows that the answer should be a number close to 40.

As with division by a single-digit number, we will sequentially move from dividing larger counting units to dividing smaller units.

The number of hundreds 8 is single-digit, so we divide 87 tens by 24. You get 3 tens and another 15 tens remain (87 - 3 24 = 15). 15 tens and 6 units is 156. And if 156 is divided by 24, you get 6 and 12 as a remainder (156 - 24 6 = 12). In total you get 3 tens and 6 units, that is, 36, and the remainder is 12. This is written like this:

10*. Find the sum of all possible two-digit numbers all of whose digits are odd.

Peterson Lyudmila Georgievna. Mathematics. 4th grade. Part 1. - M.: Yuventa Publishing House, 2005, - 64 p.: ill.

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Unfortunately, children nowadays practically do not know how to do mental calculations. This happened due to the fact that modern technologies They offer each child to solve the problem with a couple of clicks. For many children, the Internet has replaced not only textbooks, but also certain skills. You can increasingly hear from the younger generation that it is not at all necessary to know mathematics, since you always have a calculator or phone at hand. But the true significance of this science lies in the development of thinking, and not in overcoming the fear of being deceived by a trader in the market.

Long division helps students junior classes become familiar with number operations. Thanks to it, the multiplication table is fixed in memory, and the skill of performing addition and subtraction operations is honed.

To accomplish this arithmetic action You need to get acquainted with its components:

1. Dividend - a number that is divided.

2. Divisor - the number that is divided by.

3. Quotient - the result obtained by division.

4. Remainder is the part of the dividend that cannot be divided.

American and European division models

The rules for long division are the same in all countries. There is only a difference in the graphic part, that is, in its recording. IN European system The dividing line, or the so-called corner, is placed on the right side of the number being divided. The divisor is written above the corner line, and the quotient is written below the horizontal line of the corner.

Column division by American model provides for setting the corner on the left side. The quotient is written above the horizontal line of the angle, directly above the number being divided. The divisor is written under the horizontal line, to the left of the vertical line. The process of performing the action itself does not differ from the European model.

Column division by two-digit number

To use a two-digit value, you need to write it down according to the diagram, and then carry out the action. Column division begins with the highest digits of the number being divided. The first two digits are taken if the number formed by them is greater in value than the divisor. Otherwise, the first three digits are separated. The number they form is divided by the divisor, the remainder goes down, and the result is written in the dividing corner. After this, the digit from the next digit of the number being divided is transferred, and the procedure is repeated. This continues until the number is completely divided.

If it is necessary to divide a number with a remainder, then it is written separately. If you need to completely divide a number, then after the end of the digits of the number a comma is placed in the answer, indicating the beginning of the fractional part, and instead of the digits, a zero is moved down each time.

Schoolchildren learn column division, or, more correctly, the written technique of division by corner, already in the third grade. primary school, but often so little attention is paid to this topic that by 9-11 grades not all students can use it fluently. Division by a column by a two-digit number is taught in the 4th grade, as is division by a three-digit number, and then this technique is used only as an auxiliary technique when solving any equations or finding the value of an expression.

It is obvious that by paying more attention to dividing by a column than is included in school curriculum, your child will find it easier to complete math assignments up to grade 11. And for this you need little - to understand the topic and study, solve, keeping the algorithm in your head, to bring the calculation skill to automatism.

Algorithm for dividing by a two-digit number

As with division by a single-digit number, we will sequentially move from dividing larger counting units to dividing smaller units.

1. Find the first incomplete dividend. This is a number that is divided by a divisor to produce a number greater than or equal to 1. This means that the first partial dividend is always greater than the divisor. When dividing by a two-digit number, the first partial dividend must have at least 2 digits.

Examples 76 8:24. First incomplete dividend 76
265 :53 26 is less than 53, which means it is not suitable. You need to add the next number (5). The first incomplete dividend is 265.

2. Determine the number of digits in the quotient. To determine the number of digits in a quotient, you should remember that the incomplete dividend corresponds to one digit of the quotient, and all other digits of the dividend correspond to one more digit of the quotient.

Examples 768:24. The first incomplete dividend is 76. It corresponds to 1 digit of the quotient. After the first partial divisor there is one more digit. This means that the quotient will only have 2 digits.
265:53. The first incomplete dividend is 265. It will give 1 digit of the quotient. There are no more digits in the dividend. This means that the quotient will only have 1 digit.
15344:56. The first incomplete dividend is 153, and after it there are 2 more digits. This means that the quotient will only have 3 digits.

3. Find the numbers in each digit of the quotient. First, let's find the first digit of the quotient. We select an integer such that when multiplied by our divisor we get a number that is as close as possible to the first incomplete dividend. We write the quotient number under the corner, and subtract the value of the product in a column from the partial divisor. We write down the remainder. Let's check that he less than divisor.

Then we find the second digit of the quotient. We rewrite the number following the first partial divisor in the dividend into the line with the remainder. The resulting incomplete dividend is again divided by the divisor and so we find each subsequent number of the quotient until the digits of the divisor run out.

4. Find the remainder(if there is).

If the digits of the quotient run out and the remainder is 0, then the division is performed without a remainder. Otherwise, the quotient value is written with a remainder.

Division by any multi-digit number (three-digit, four-digit, etc.) is also performed.

Analysis of examples of dividing by a column by a two-digit number

First, let's look at simple cases of division, when the quotient results in a single-digit number.

Let's find the value of the quotient numbers 265 and 53.

The first incomplete dividend is 265. There are no more digits in the dividend. This means that the quotient will have a single digit number.

To make it easier to choose the quotient number, let's divide 265 not by 53, but by a close round number 50. To do this, divide 265 by 10, the result will be 26 (the remainder is 5). And divide 26 by 5, there will be 5 (remainder 1). The number 5 cannot be immediately written down in the quotient, since it is a trial number. First you need to check if it fits. Let's multiply 53*5=265. We see that the number 5 has come up. And now we can write it down in a private corner. 265-265=0. The division is completed without remainder.

The quotient of 265 and 53 is 5.

Sometimes when dividing, the test digit of the quotient does not fit, and then it needs to be changed.

Let's find the value of the quotient numbers 184 and 23.

The quotient will be a single digit number.

To make it easier to choose the quotient number, let's divide 184 not by 23, but by 20. To do this, divide 184 by 10, there will be 18 (remainder 4). And we divide 18 by 2, the result is 9. 9 is a test number, we won’t immediately write it in the quotient, but we’ll check if it’s suitable. Let's multiply 23*9=207. 207 is greater than 184. We see that the number 9 is not suitable. The quotient will be less than 9. Let's try to see if the number 8 is suitable. Let's multiply 23*8=184. We see that the number 8 is suitable. We can write it down privately. 184-184=0. The division is completed without remainder.

The quotient of 184 and 23 is 8.

Let's consider more complex cases division.

Let's find the value of the quotient of 768 and 24.

The first incomplete dividend is 76 tens. This means that the quotient will have 2 digits.

Let's determine the first digit of the quotient. Let's divide 76 by 24. To make it easier to choose the quotient number, let's divide 76 not by 24, but by 20. That is, you need to divide 76 by 10, there will be 7 (the remainder is 6). And divide 7 by 2, you get 3 (remainder 1). 3 is the test digit of the quotient. First let's check if it fits. Let's multiply 24*3=72. 76-72=4. The remainder is less than the divisor. This means that the number 3 is suitable and now we can write it in place of the tens of the quotient. We write 72 under the first incomplete dividend, put a minus sign between them, and write the remainder under the line.

Let's continue the division. Let's rewrite the number 8 following the first incomplete dividend into the line with the remainder. We get the following incomplete dividend – 48 units. Let's divide 48 by 24. To make it easier to find the quotient, let's divide 48 not by 24, but by 20. That is, if we divide 48 by 10, there will be 4 (the remainder is 8). And we divide 4 by 2, it becomes 2. This is the test digit of the quotient. We must first check if it will fit. Let's multiply 24*2=48. We see that the number 2 fits and, therefore, we can write it in place of the units of the quotient. 48-48=0, division is performed without remainder.

The quotient of 768 and 24 is 32.

Let's find the value of the quotient numbers 15344 and 56.

The first incomplete dividend is 153 hundreds, which means that the quotient will have three digits.

Let's determine the first digit of the quotient. Let's divide 153 by 56. To make it easier to find the quotient, let's divide 153 not by 56, but by 50. To do this, divide 153 by 10, the result will be 15 (remainder 3). And we divide 15 by 5, it becomes 3. 3 is the test digit of the quotient. Remember: you cannot immediately write it down in private, but you must first check whether it is suitable. Let's multiply 56*3=168. 168 is greater than 153. This means that the quotient will be less than 3. Let’s check if the number 2 is suitable. Multiply 56*2=112. 153-112=41. The remainder is less than the divisor, which means that the number 2 is suitable, it can be written in the place of hundreds in the quotient.

Let us form the following incomplete dividend. 153-112=41. We rewrite the number 4 following the first incomplete dividend into the same line. We get the second incomplete dividend of 414 tens. Let's divide 414 by 56. To make it more convenient to choose the quotient number, let's divide 414 not by 56, but by 50. 414:10=41(rest.4). 41:5=8(rest.1). Remember: 8 is a test number. Let's check it out. 56*8=448. 448 is greater than 414, which means that the quotient will be less than 8. Let's check if the number 7 is suitable. Multiply 56 by 7, we get 392. 414-392=22. The remainder is less than the divisor. This means that the number fits and in the quotient we can write 7 in place of tens.

We write 4 units in the line with the new remainder. This means the next incomplete dividend is 224 units. Let's continue the division. Divide 224 by 56. To make it easier to find the quotient number, divide 224 by 50. That is, first by 10, there will be 22 (the remainder is 4). And divide 22 by 5, there will be 4 (remainder 2). 4 is a test number, let’s check it to see if it’s suitable. 56*4=224. And we see that the number has come up. Let's write 4 in place of units in the quotient. 224-224=0, division is performed without remainder.

The quotient of 15344 and 56 is 274.

Example for division with remainder

To make an analogy, let's take an example similar to the example above, and differing only in the last digit

Let's find the value of the quotient 15345:56

We first divide in the same way as in the example 15344:56, until we reach the last incomplete dividend 225. Divide 225 by 56. To make it easier to choose the quotient number, divide 225 by 50. That is, first by 10, there will be 22 (the remainder is 5 ). And divide 22 by 5, there will be 4 (remainder 2). 4 is a test number, let’s check it to see if it’s suitable. 56*4=224. And we see that the number has come up. Let's write 4 in place of units in the quotient. 225-224=1, division done with remainder.

The quotient of 15345 and 56 is 274 (remainder 1).

Division with zero in quotient

Sometimes in a quotient one of the numbers turns out to be 0, and children often miss it, hence the wrong solution. Let's look at where 0 can come from and how not to forget it.

Let's find the value of the quotient 2870:14

The first incomplete dividend is 28 hundreds. This means that the quotient will have 3 digits. Place three dots under the corner. This important point. If a child loses a zero, there will be an extra dot left, which will make them think that a number is missing somewhere.

Let's determine the first digit of the quotient. Let's divide 28 by 14. By selection we get 2. Let's check if the number 2 fits. Multiply 14*2=28. The number 2 is suitable; it can be written in place of hundreds in the quotient. 28-28=0.

The result was a zero remainder. We've marked it in pink for clarity, but you don't need to write it down. We rewrite the number 7 from the dividend into the line with the remainder. But 7 is not divisible by 14 to obtain an integer, so we write 0 in the place of tens in the quotient.

Now we rewrite the last digit of the dividend (number of units) into the same line.

70:14=5 We write the number 5 instead of the last point in the quotient. 70-70=0. There is no remainder.

The quotient of 2870 and 14 is 205.

Division must be checked by multiplication.

Division examples for self-test

Find the first incomplete dividend and determine the number of digits in the quotient.

3432:66 2450:98 15145:65 18354:42 17323:17

You have mastered the topic, now practice solving several examples in a column yourself.

1428: 42 30296: 56 254415: 35 16514: 718

Division multi-digit or multi-digit numbers are convenient to produce in writing in a column. Let's figure out how to do this. Let's start by dividing a multi-digit number by a single-digit number, and gradually increase the digit of the dividend.

So let's divide 354 on 2 . First, let's place these numbers as shown in the figure:

We place the dividend on the left, the divisor on the right, and the quotient will be written under the divisor.

Now we begin to divide the dividend by the divisor bitwise from left to right. We find first incomplete dividend, to do this, take the first digit on the left, in our case 3, and compare it with the divisor.

3 more 2 , Means 3 and there is an incomplete dividend. We put a dot in the quotient and determine how many more digits will be in the quotient - the same number as remained in the dividend after selecting the incomplete dividend. In our case, the quotient has the same number of digits as the dividend, that is, the most significant digit will be hundreds:

In order to 3 divide by 2 remember the multiplication table by 2 and find the number, when multiplied by 2 we get the greatest product, which is less than 3.

2 × 1 = 2 (2< 3)

2 × 2 = 4 (4 > 3)

2 less 3 , A 4 more, which means we take the first example and the multiplier 1 .

Recording 1 to the quotient in place of the first point (in the hundreds place), and write the found product under the dividend:

Now we find the difference between the first incomplete dividend and the product of the found quotient and the divisor:

The resulting value is compared with the divisor. 15 more 2 , which means we have found the second incomplete dividend. To find the result of division 15 on 2 again remember the multiplication table 2 and find the greatest product that is less 15 :

2 × 7 = 14 (14< 15)

2 × 8 = 16 (16 > 15)

The required multiplier 7 , we write it as a quotient in place of the second point (in tens). We find the difference between the second incomplete dividend and the product of the found quotient and divisor:

We continue the division, why we find third incomplete dividend. We lower the next digit of the dividend:

We divide the incomplete dividend by 2, and put the resulting value in the quotient units category. Let's check the correctness of the division:

2 × 7 = 14

We write the result of dividing the third incomplete dividend by the divisor into the quotient and find the difference:

We got the difference equal to zero, which means the division is done Right.

Let's complicate the task and give another example:

1020 ÷ 5

Let's write our example in a column and define the first incomplete quotient:

The thousands place of the dividend is 1 , compare with the divisor:

1 < 5

We add the hundreds place to the incomplete dividend and compare:

10 > 5 – we have found an incomplete dividend.

We divide 10 on 5 , we get 2 , write the result into the quotient. The difference between the incomplete dividend and the result of multiplying the divisor and the found quotient.

10 – 10 = 0

0 we do not write, we omit the next digit of the dividend – the tens digit:

We compare the second incomplete dividend with the divisor.

2 < 5

We should add one more digit to the incomplete dividend; for this we put in the quotient, on the tens digit 0 :

20 ÷ 5 = 4

We write the answer in the category of units of the quotient and check: we write the product under the second incomplete dividend and calculate the difference. We get 0 , Means example solved correctly.

And 2 more rules for dividing into a column:

1. If the dividend and divisor have zeros in the low-order digits, then before dividing they can be reduced, for example:

As many zeros in the low-order digit of the dividend we remove, we remove the same number of zeros in the low-order digits of the divisor.

2. If there are zeros left in the dividend after division, then they should be transferred to the quotient:

So, let’s formulate the sequence of actions when dividing into a column.

1. Place the dividend on the left and the divisor on the right. We remember that we divide the dividend by isolating incomplete dividends bit by bit and dividing them sequentially by the divisor. The digits in the incomplete dividend are allocated from left to right from high to low.
2. If the dividend and divisor have zeros in the lower digits, then they can be reduced before dividing.
3. We determine the first incomplete divisor:

A) select the highest digit of the dividend into the incomplete divisor;

b) compare the incomplete dividend with the divisor; if the divisor is larger, then go to point (V), if less, then we have found an incomplete dividend and can move on to point 4 ;

V) add the next digit to the incomplete dividend and go to point (b).

1. We determine how many digits there will be in the quotient, and put as many dots in place of the quotient (under the divisor) as there will be digits in it. One point (one digit) for the entire first incomplete dividend and the remaining points (digits) are the same as the number of digits left in the dividend after selecting the incomplete dividend.
2. We divide the incomplete dividend by the divisor; to do this, we find a number that, when multiplied by the divisor, would result in a number either equal to the incomplete dividend or less than it.
3. We write the found number in place of the next quotient digit (dot), and write the result of multiplying it by the divisor under the incomplete dividend and find their difference.
4. If the difference found is less than or equal to the incomplete dividend, then we have correctly divided the incomplete dividend by the divisor.
5. If there are still digits left in the dividend, then we continue division, otherwise we go to point 10 .
6. We lower the next digit of the dividend to the difference and get the next incomplete dividend:

a) compare the incomplete dividend with the divisor, if the divisor is greater, then go to point (b), if less, then we have found the incomplete dividend and can proceed to point 4;

b) add the next digit of the dividend to the incomplete dividend, and write 0 in the place of the next digit (dot) in the quotient;

c) go to point (a).

10. If we performed division without a remainder and the last difference found is equal to 0 , then we did the division correctly.

We talked about dividing a multi-digit number by a single-digit number. In the case where the divider is larger, division is performed in the same way:

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 breakingdividing by a series of more simple 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 with 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: than more difference in the number of characters in the entries of the dividend and divisor, the morespace 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 next point 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. Translation decimals in ordinary and ordinary 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.

• We 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 upon division natural number to 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.