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Division

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Introduction: If the divisor, divisor, quotient and remainder are respectively a, b, q and r then a = b × q + r

Example: On the number divided by 96, 47 remains. If the same number is divided by 32, then what will be the remaining?
Solution:
a = 96 × q + 47 = 3q × 32 + 1 × 32 + 15 = (3q + 1) × 32 + 15

Therefore, remaining part of the same divide by 32, 15 will survive.

Note: If the second denominator is first a factor of the denominator, then the remainder is determined by dividing the other divisor in the remainder to find the second remainder, but if the first is smaller than the second divisor then the second remainder It is the same (the first remainder).


 

Section – A

Testing the divisibility

* Condition to be divided by 2 – If the number of a unit of an entity is zero (0) or divisible by 2, then that number will be divisible by 2.
E.g., 7530, 8436, 4328 etc.

 

* Condition to be divided by 3 – If the sum of digits present in any number is divisible by 3, then that number will be completely divided by 3.
E.g. 747, 2343, 6273 etc.

∵ 7 + 4 + 7 = 12 which is divisible by ‘3’. Therefore 747 3 is the complete number divided. Similarly the sum of the digits of other numbers is also divisible by 3 ‘. Hence all numbers are numbers that are completely divided by ‘3’.

 

* Condition to be divided by 4 – If the last two digits of a number (tens and unit digits) are ’00’ or divisible by 4, that number will be divided by 4.
Eg -74300, 15712, 4308 etc.
00, 12 and 08  are divisible by ‘4’ . Therefore the above numbers will be divided by 4.

 

* Condition to be divided by 5 – If the number of a unit of a unit is zero (0) or 5, then that number will be divided by 5. Eg- 930, 5465, 8745 etc.

 

* 6 If the sum of points of any number is divisible by 3, then it will be divided entirely by number 6. Eg -3564, 1572, 4248 etc.

 

* Condition to be divided by 3, 7, 11, 13, 21 and 37 –

(i) The smallest number formed from the recurrence of a six digit number or number will be divided by 3, 7, 11, 13, 21 and 37 of any number. E.g., 222222, 444444, 323232323232 and so on.
Note: If the number is repeated in the multiplication of 6 such as -6 times, 12 times, 18 times or 24 times, then the numbers 3, 7, 11, 13, 21 or 37 Will be completely divided by any number.

 

(ii) If the fourth digit in the number of four digits is subtracted from the first digit by subtracting the balance from the first digit, then the remainder will be divided by the original number seven. For example, 6545 is divisible by 7, because (6 – 5) 54 means 154, is divisible by 7.

 

(iii) If the number obtained by reducing the fifth digit of a number with five digits is reduced from the number of two digits, then the number 7 will be divided if the number is divisible by 7. -Now 23632 is divisible by 7, because (23 -2) 63 is variant from 2163 point 7.

 

(iv) If a number is divided into three groups of three and the sum of the groups located on odd places and the sum of groups located at even places If the difference is zero (0) or divisible by 7, then that number will be divided by 7. As if -266133 divided into two groups of three-three digits, respectively, the group 266 at the odd position and the group at the other place is 133. Since 266 – 133 = 133, the digit is divisible by 7. Therefore the said number will be divided by 7.

 

* Condition to be divided by 8 – If the last three digits of a number (hundredth, tenth, and unit number) are ‘000 or divisible by 8, then the number will divide by 8. Like -347000, 67512, 954024 etc.
000, 512 and 024 are divisible by 8 numbers, so the above numbers are divisible by 8.
* Condition to be divided by 9 – If the sum of a number digits is divisible by 9, then that number is divisible by 9.
The sum of digits like -3458, 75438, 5432787, etc. is equal to 9. Therefore these numbers are divisible by 9.

 

* Condition to be divided by 10 – If the number of a unit of a unit is zero (0), then the number is divisible by 10. –

 

* Condition to be divided by 11 – If the sum of digits present at the odd places of a number and the sum of the digits present at the points is multiplied or 11 is multiplied by 11, then Will split up.
As-54285, the difference between the digits 5, 2 and 5 of the odd places 12 and the sum of the digits 4 and the sum of 12 is zero (0). Therefore, this number is divisible by 11.

 

* Condition to be divisible by 12 – If the sum of digits of a number is divisible by 3 and the number of the last two digits of the number (ten and the digit of the number) is divisible by 4, then that number Will be divisible by 12.

The sum of digits of 94536, 7824, 23424 etc. is divisible to 3 and their last two digits are divisible by 4. Therefore these numbers will be divisible from 12.

 

* Condition to be divisible by 14 – If any number is divisible by 7, that number will also be divided by 14. Such as -444444, 666666, 36652 etc.

 

* Condition to be divisible by 15 – If the number of a unit of a unit is 0 or 5 and the sum of its digits is divisible by 3, then the number is divisible by 15. Eg -78510, 6645, 6795 etc.

 

* Condition to be divisible by 16 – If the last four digits of a number are ‘0000 or divisible by 16, then that number will be divided by 16. Eg -3450000, 231968, 745136 etc.

 

* Condition to be divisible by 18 – If the sum of any numbers is divisible by 9, then that number will be divided by 9. Eg-3582, 8874, 8658 etc.

 

Section – B

GENERAL METHOD OF DIVISION ON WHOLE

Division by a two-digit divisor:
Example
8262 = 34?
Solution:

1st step. Firstly the decimal point has been placed before the unit number ‘2’ of the division.

2nd step. In the first term of the division, ‘8’ which is also the first term of AD, dividing the ten digit ‘3’ of the divisor by dividing the remaining remainder ‘2’ before the second post of the second. has gone. I.e. PD = 22

3rd step . The second part of the factorial is PD made of 2. Out of 22 the first digit of the quotient ‘2’ and the denominator of the denominator unit number 4, the number obtained is AD. Has been placed at the place of second position. I.e. (22-2 × 4) = 14

4th step . A.D. In the second term ’14’, dividing the ten digits of the divisor by ‘3’, the remainder ‘2’ has been placed before the third term ‘6’ of the work. I.e., PD = 26

5th step . The third term of the work is made by P.D. The second digit of the quotient from 26 and the number of divisor unit number 4 decreasing the number obtained by A.D. Has been placed in place of the third position. I.e. (26-4 × 4) = 10

6th step . A.D. In the third verse 10, dividing the ten digits of the divisor by dividing it from ‘3’, the remainder ‘1’ has been placed before the fourth digit 2 of the factorial. I.e. PD = 12

7th step . P.D. The third digit of the quotient from 12 and the number of divisor unit number 4 decreasing the number obtained by A.D. Has been placed at the place of fourth position.
That is (12-4 × 3) = 0

Last step . A.D. In the fourth term “0”, dividing the tenth digit of the divisor by 3 divided by the quotient is given after the decimals in the desired quotient of ‘0’. Thus the desired quotient = 243.

Note: * P.D. (Provisional Dividend): The number obtained by keeping the first, second, and third remainder before the second, third and fourth digits of the semester, respectively, is P.D. Is called.
* A.D. (Actual Dividend):
In which the dividend quotient is divided by dividing the digits A.D. Is called.

 

Example: 10395 ÷ 45?

1st Step. First, the decimal has been placed before the last one digit of the division.

2nd Step. 10/4 ⇒ quotient = 2 and remainder = 2

3rd Step. (23 – 5 × 2) = (23-10) = 13

4th step. 13/4 ⇒ quotient = 3 and remainder = 1

5th Step . (19 – 5 × 3) = (19-15) = 4

6th Step. 4/4 ⇒ quotient = 1 and remainder = 0

7th Step. (05 – 5 × 1) = (5-5) = 0

Last Step. 0/4 = 0,
Therefore, the expected quotient = 231.

 

Exercise ⇒ Solve them by muscular action

1. 1701 ÷ 21 =?
2. 108489 ÷ 43 =?
3. 38982 ÷ 73 =?
4. 18939 ÷ 59 =?
5. 128094 = 37?

 

Division by a three-digit divisor:

Example: 218064 ÷ 413 =?
Solution:

1st step. First, the decimal is placed after the two digits on the right side of the division.

2nd step. The first post of the dividend 21 is A.D. Is also the first post, divided by dividing the number of divisor by 4 divided by 4 divided by the second term ‘8’. I.e. PD = 18

3rd step. PD’s made of second term ‘8’ The number obtained by dividing the first digit 5 ​​of the quotient from the ’18’ and the denominator of the divisor’s tenth digit 1 is A.D. Has been placed at the place of second position. I.e. (18-5 × 1) = 13

4th step. A.D. In the second verse 13, dividing the hundredth digit of the divisor by 4 divided by 4 divided by 5, the third term of the divisement was placed before ‘0’. I.e. P.D. = 50

5th step. The third position of the division is PD 0. The number obtained by reducing the sum of the product obtained by multiplying the first digit of the quotient from ’50 and the second digit 2 by multiplying by the unit number 3 of the denominator and the tenth digit ‘1’ respectively. Has been placed in place of the third position. That is 50 ー (5 × 3 + 2 × 1) = (50-17) = 33

6th step. A.D. The third term of ’33 ‘divided by dividing the hundredth digit of the denominator 4, the remainder’ 1 ‘has been placed before the fourth term’ 6 ‘of the work i.e. PD = 16.

7th step. The PD made the fourth term 6 of the factorial. In the number of “16”, the second digit of the quotient ‘2’ and the third digit ‘8’, respectively, by dividing the unit number 3 of the divisor and the tenth digit by 1, decreasing the sum of the product obtained by A.D. Has been placed at the place of fourth position. That is {16- (2 × 3 + 8 × 1)} = (16 – 14) = 2

8th step. A.D. In the fourth verse of 2, the number of denominator of the divisor does not run from the number 4 because in the quotient, a zero (0) has been placed after the decimal.

9th Step. The third point of AD’s ‘8’ and the fourth digit ‘0’ is the unit number 3 of the divisor, and the tenth digit ‘1’, the PD, the fourth term ‘AD’ of the first ‘4’ ‘The number obtained by reducing the sum of the product obtained by multiplying’ Has been placed at the place of fifth position. That is 24- (8 × 3 + 0 × 1) = (24-24) = 0

Last step. By dividing the fifth term ‘0’ of AD with the hundredth digit of the divisor ‘4’, the quotient quotient of 0 is placed in the second place after the decimal in the desired quotient. Thus the desired quotient = 528 Ans.

 

Example : 135785 ÷ 2089 =?
Solution :

1st Step . First, the decimal is given before the last two digits of the division.

2nd Step. 135/20 ⇒ quotient = 6 and the remaining = 15

3rd Step. 157 – (6 × 8) = 157-48 = 109

4th Step. 109/20 ⇒ quotient = 5 and balance = 9

5th Step. 98 – (8 × 5 + 9 × 6) = 98-94 = 4

6th Step. 45- (8 × 0 + 9 × 5) = 45-45 = 0
Last Step.
0/20

So the expected quotient = 65 Ans.

 

Division by a four-digit divisor:

Example 3739794 ÷ 5123 =?
Solution :

1st step. First, the decimal point has been placed after the three digits on the right side of the division.

2nd step. The first post of the dividend 37 is A.D. Is also the first term, in which the remaining ‘2’ received by dividing the number of divisor by 15 divided by 15 is placed before the second term of the work ‘3. I.e. PD = 23

3rd step. The second part of the second part, ‘PD’ 3, the first digit of the quotient from 23 and the number of divisor number 1, decreasing the product obtained by A.D. Has been placed at the place of second position. I.e. (23-7 × 1) = 16

4th Step. In the second term ‘AD’ of AD, the remainder of the part obtained by dividing the number of divisor by ‘5’ has been left before the third term ‘9’ of the work. I.e. PD = 19

5th step. P.D. = ‘The first digit of the quotient from’ 19 ‘and the second digit 3 is respectively multiplied by the ten digits of the divisor,’ 2 ‘and the number of hundredths,’ 1 ‘, decreasing the sum of the product obtained by A.D. Has been placed in place of the third position. That is, {19 – (7 × 2 + 3 × 1)} = (19 – 17) = 2

6th step. A.D. In the third verse 2, the number of denominator of the denominator is not part of the number 5 because in the quotient, one (0) is placed before the decimal after the decimal. I.e. PD = 27

7th step. P.D. The first digit of the quotient from 27, the second digit 3 and the third digit ‘0’, respectively, by dividing the unit number 3, the tenth digit ‘2’ and the number of hundredths by 1, 0 ‘to AD Has been placed at the place of fourth position. That is, {(27) – (7 × 3 + 3 × 2 + 0 × 1)} = (27-27) = 0

8th step. A.D. In the fourth term ‘0’, dividing the number of the denominator by dividing the number 5, the quotient obtained is zero (0) in the quotient after the decimal place and the second place is A.D. The fourth term of ‘0’ has been placed before the fifth term of the work ‘9’ i.e. PD = 09.

9th step. P.D. Lowering the sum of the product obtained by dividing the second digit of the quotient 3, the third digit ‘0’ and the fourth digit ‘0’, respectively, by dividing the unit number 3, the tenth digit ‘2’ and the number of hundredths by ‘1’. The given number ‘0’ is AD Has been placed at the place of fifth position. That is 109 – (3 × 3 + 0 × 2 + 0 × 1) = (9-9) = 0

Last step. In this final post ‘AD’ of the divisor by dividing the number of the denominator by ‘5’, the quotient quotient is zero (0) placed in the third position after the decimal in the desired quotient. Thus the expected quotient = 73 Ans.

 

Example: 3614.34 ÷ 6341 =?

1st step . In the first place, the decimal is placed in the decimal point by three points and left side.

2nd Step . 3.6 / 6 ⇒ quotient = 0.5 and remainder = 6

3rd Step . P.D. = (61 – 3 × 5) = (61 – 15) = 46

4th step . 46/6 ⇒ quotient = 7 and remainder = 4,

5th Step . P.D. = {44 – (3 × 7 + 4 × 5)} = (44-41) = 3

6th step . 3 will not run from 6 Therefore, a zero (0) has been placed in the quotient.

7th Step . PD = {33-0 × 3 + 7 × 4 + 5 × 1}} = (33-33) = 0

8th step . 0/6 = 0 This quotient is zero (0) sitting in the desired quotient.

9th Step . P. D. = 07 – (0 × 3 + 0 × 4 + 7 × 1) = (7-7) = 0

Last step. 0/6 = 0, this quotient zero (0) has also been sat in the desired quotient.

Hence expected quotient = 0.57 Ans.

 

Exercise – solve them with monthly action

1. 276544 ÷ 6321 =?
2. 200651 ÷ 54.32 =?
3. 204183 ÷ 3241 =?
4. 488.29 ÷ 2123 -?
5. 2281.44 ÷ 5432 =?

 

Section-C

SPECIAL METHOD of DIVISION on WHOLE NUMBERS

Division by 5 25 125 and 625

TRICK → If any number is to be divided by 5, 25, 125 or 625, the number obtained by multiplying the number by 2, 4, 8, or 16, respectively, respectively, from the right one in the product The first decimal is placed before the two, three or four digits i.e. the quotient is obtained by dividing 10, 100, 1000 and 10000 respectively in the product.

 

Division by 75

TRICK → If a number is to be divided by 75, then the quotient divided by 3 divided by 3 is multiplied by 4. After achieving the first decimation of decimal in the given product, the expected quotient is obtained.

 

Solve Exercise → with these TRICK

1. 57632 ÷5 =?
2. 32546 ÷25 =?
3. 74.3253 ÷ 125 =?
4. 23457 ÷ 625 =?
5. 546.372 ÷ 75 =?

 

 

 

Division by 70, 300, 400, 7000, 9000 etc.

TRICK
(i)
First, the number of zero (0) in the denominator is divided by the remaining number of the denominator (except zero) except the number of the number on the right side of the factorial.

(ii) Thus leaving the remainder left to the left of the divisive part of the divisive part gives the desired remainder.

Exercise → Solve them by TRICK

1. 95432 ÷60 =?
2. 234684 ÷ 400 =?
3. 468321 ÷ 700 =?
4. 875432 ÷ 9000 =?
5. 534687 + 3000 =?

 

Division by 45, 350, 4500 etc:

TRICK
(i)
Firstly the dividend and divisor are multiplied by 2. The number of zero (0) on the right in the divisor obtained after multiplying

(ii)  after multiplying, except for that number, the remaining number of the new divisor (except zero).

(iii) Thus leaving the remaining remainder of the new dividend left to the left, the desired remainder is achieved.

(iv) The ½ part of this intended remainder is known.

Exercise → Solve them by TRICK

1. 42346 ÷ 150 =?
2. 75432 ÷ 450 =?
3. 87543 = 2500?
4. 324563 ÷ 6500 =?
5. 754362 ÷ 4500 =?

 

Division by 9 99 999 9999 etc

Action Method

(i) In the first case, the number of times the number of repeat marks in the divisor is 9, the number of the same number is divided by the number of the part of the division, if the left is 9 If he survives less than the number of times, then he is also kept as the last group.

(ii) Now a vertical streak is drawn to find partiality and remainder.

(iii) On the right hand side of this vertical streak, the digits of a group on the right-most of the points of factorial are placed and the remaining number is placed on the left side of the streak.

(iv) Then the second group is placed on the right and right side of the ration and the remaining number is placed on the left side of the line.

(v) Similarly, the last group is also left to the left.

(vi) All groups on the right side are added. If the sum is smaller than the divisor then the remainder remains the same, but if the sum is more then the number of the leftmost points is taken as a gain. If you add this achievement to the remaining number of the sum, then the desired remainder is achieved.

(vii) In the sum of the groups on the left side, a surefire quotient is achieved when adding the gain achieved from the sum of the groups on the right.

 

 

Example 1 . 53246 ÷ 9 =?

Solution:

Quotient = (5914 + 2) = 5916 and remainder = 0 + 2 = 2.

? = 5916  2Ans.

 

Example 2 . 453267 ÷ 99 =?

Solution:

Quotient = (4577 + 1) = 4578 and remainder = (44 + 1) = 45

4578 4599 = 4578 511  Ans.

 

Example 3. 4378546 ÷ 999 = ?
Solution:

Quotient = 4382 and remainder = 928

? =4382 = 928999 Ans.

 

Example 4. 2345746 ÷ 9999 = ?
Solution:

Quotient = 234 and remainder = 5977

? = 234 59779999 Ans.

 

Exercise → Solve them by TRICK

1. 3257 ÷ 9 =?
2. 457632 ÷ 99 =?
3. 8532432 ÷ 999 =?
4. 23456782 ÷ 9999 =?
5. 61234632 ÷ 99999 =?

 

Division by any number small and in the neighbourhood of 100 1,000 or 10,000

Action Method

(i) First, a vertical streak is drawn to find the quotient and remainder.

(ii) On the right side of this perpendicular line, the number of points in the denominator is kept as the number of digits from the point of partiality.

(iii) Then the number on the left side of the streak is multiplied by the number as it is, the denominator decreases in divisor from 100, 1000 or 10,000. While writing this product, it is also kept in mind that its unit number is in front of the unit number of the first group.

(iv) The process of multiplying and writing down the number of decreases is done until a single digit is left to the left of the line.

(v) The numbers on the right side are added. If the sum is smaller than the divisor then the remainder is the same but if the sum is more then the number of the leftmost points is taken as a gain. By multiplying this number by the number of points achieved by this, the remaining remainder of the sum of the sum is achieved.

(vi) In the sum of the numbers on the left hand, a positive quotient is gained when the gain obtained from the sum of the numbers on the right side.

 

Example. 23465 ÷ 92 = ?

Solution :

1st step .
2nd Step . (234 × 8)
3rd Step . (18 × 8)
4th Step . (1 × 8)

Quotient = (253 + 1) = 254 and remainder = (89 + 1 × 8) = 97 (which is greater than the denominator 92)Expected remainder = (97-92) = 5 and expected quotient = (254 + 1) = 255
? = 255 592 Ans.

Note – If the value of the remainder comes more than the denominator, then in such a situation, the remainder of the remainder is achieved after reducing the denominator from that remainder. The value of the remainder is equal to the number of times greater than the divisor, adding a greater multiplication to the quotient leads to the quotient expected.

 

Example 3. 34532 ÷ 991 =?
Solution:

1st step .
2nd Step . (34 × 9)

Quotient = 34 and remainder = 838

?=34 838991 Ans.

 

Example 4. 2354647 ÷ 9996 =?
Solution :

1st Step.

2nd Step. (235 × 4)

Quotient = 235 and remainder = 5587

? = 235  55879996 Ans.

 

Exercise → Solve them by TRICK

1. 43267 ÷ 94 =?
2. 523748 ÷ 992 =?
3. 7432165 ÷ 9988 =?
4. 81234687 ÷ 9993 =?
5. 321468753 ÷ 99991 =?
Division by continuing different divisors

TRICK →

 

Division of mixed numbers by whole numbers or fractions.

Division by whole number

TRICK →

 

Division by Fractions

TRICK

 

Division of Decimal: While dividing the decimals into numbers, the first is to remove the decimal from the divisor and divisor and make the whole number. After this the quotient is obtained by dividing the divisor into the divisor.

Case 1 . If the number of digits in decimal and the number of digits in the divisor is greater than the number of digits on the right of the decimal, then the decimal is given before the digits on the right side of the quotient equal to the difference of the difference.

Case 2 . If the divisor is given the right and right of the decimal and the number of digits in the divisor, the right and zero of the sarbambar quotient in the divisor.

Here, in the factorial, the total number of digits on the right of the decimal of the decimal = 8 and the number of points on the right of the decimal in the denominator = 5
Their difference = (8 – 5) = 3
∴? = 1.008 Ans.

Here, in the factorial, the total number of digits on the right of decimal = 7 and
In the denominator, the total number of points on the right of decimal = 13
Their difference = (13-7) = 6
∴? = 40, 50, 00,000 Ans.

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