Tips And Tricks For Speedy Calculations Module 4 Division

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TIPS AND TRICKS FOR SPEEDY CALCULATIONS – MODULE IV – DIVISION

Division is no more tedious as it used to be! In this module we deal with divisibility rules of various numbers, how to find remainders and general division rules. After reading this module and solving problems by heart you may find division as easier as multiplication.

 

  1. Simple divisibility rules by numbers such as 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 16

To check whether a number is divisible by the above numbers or not either you can perform the actual division or use the pragmatic short cuts given. We recommend you to use these short cuts as these are much simpler than the long and conventional division methods.

Number

Method

3

If the sum of its digits is divisible by 3

4

If the number formed by the last two digits is divisible by 4

6

If it is divisible by both 2 & 3.

7

If after subtraction of a number consisting of the last three digits from a number consisting of the rest of its digits the result is a number that can be divisible by 7. Ex.: 414141 is divisible 7 as 414-141= 273 is divisible by 7.

8

If the last three digits of the number are divisible by 8

9

If the sum of its digit is divisible by 9.

11

If the difference of the sum of its digits at odd places and sum of its digits at even places, is either 0 or a number divisible by 11. Ex: 4832718 is divisible by 11, since:
(Sum of digits at odd places) – (sum of digits at even places)
= (8+7+3+4)-(1+2+8) = 11

12

A number is divisible by 12 if it is divisible by both 4 and 3

14

If a number is divisible by 2 as well as 7

15

If a number is divisible by both 3 & 5

16

If the number formed by the last 4 digits is divisible by 16

 

 

  1. Advanced divisibility rules for 7, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47

 

These are must to be known for beginning and building your grip on division as a full mathematical operation. The following rules explains whether test for a number to be fully divisible by the above mentioned numbers.

Number

Method

Example

7

Subtract 2 times the last digit from remaining truncated number. Repeat the step as necessary. If the result is divisible by 7, the original number is also divisible by 7

Check for 945: : 94-(2*5)=84. Since 84 is divisible by 7, the original no. 945 is also divisible

13

Add 4 times the last digit to the remaining truncated number. Repeat the step as necessary. If the result is divisible by 13, the original number is also divisible by 13

Check for 3146:: 314+ (4*6) = 338:: 33+(4*8) = 65. Since 65 is divisible by 13, the original no. 3146 is also divisible

17

Subtract 5 times the last digit from remaining truncated number. Repeat the step as necessary. If the result is divisible by 17, the original number is also divisible by 17

Check for 2278:: 227-(5*8)=187. Since 187 is divisible by 17, the original number 2278 is also divisible.

19

Add 2 times the last digit to the remaining truncated number. Repeat the step as necessary. If the result is divisible by 19, the original number is also divisible by 19

Check for 11343:: 1134+(2*3)= 1140. (Ignore the 0):: 11+(2*4) = 19. Since 19 is divisible by 19, original no. 11343 is also divisible

23

Add 7 times the last digit to the remaining truncated number. Repeat the step as necessary. If the result is divisible by 23, the original number is also divisible by 23

Check for 53935:: 5393+(7*5) = 5428 :: 542+(7*8)= 598:: 59+ (7*8)=115, which is 5 times 23. Hence 53935 is divisible by 23

29

Add 3 times the last digit to the remaining truncated number. Repeat the step as necessary. If the result is divisible by 29, the original number is also divisible by 29

Check for 12528:: 1252+(3*8)= 1276 :: 127+(3*6)= 145:: 14+ (3*5)=29, which is divisible by 29. So 12528 is divisible by 29

31

Subtract 3 times the last digit from remaining truncated number. Repeat the step as necessary. If the result is divisible by 31, the original number is also divisible by 31

Check for 49507:: 4950-(3*7)=4929 :: 492-(3*9) :: 465:: 46-(3*5)=31. Hence 49507 is divisible by 31

37

Subtract 11 times the last digit from remaining truncated number. Repeat the step as necessary. If the result is divisible by 37, the original number is also divisible by 37

Check for 11026:: 1102 – (11*6) =1036. Since 103 – (11*6) =37 is divisible by 37. Hence 11026 is divisible by 37

41

Subtract 4 times the last digit from remaining truncated number. Repeat the step as necessary. If the result is divisible by 41, the original number is also divisible by 41

Check for 14145:: 1414 – (4*5) =1394. Since 139 – (4*4) =123 is divisible by 41. Hence 14145 is divisible by 41

43

Add 13 times the last digit to the remaining truncated number. Repeat the step as necessary. If the result is divisible by 43, the original number is also divisible by 43.

Check for 11739:: 1173+(13*9)= 1290:: 129 is divisible by 43. 0 is ignored. So 11739 is divisible by 43

47

Subtract 14 times the last digit from remaining truncated number. Repeat the step as necessary. If the result is divisible by 47, the original number is also divisible by 47.

Check for 45026:: 4502 – (14*6) =4418. Since 441 – (14*8) =329, which is 7 times 47. Hence 45026 is divisible by 47

 

  1. Calculating remainder when a very large number is divided by a smaller one.

 

The statement- “When x is divided by z, it leaves y as the remainder.” is represented in modular arithmetic as x=y(mod z)

It can also be interpreted as “x and y leave the same remainder when divided by z.” This is also known as the congruence relation and we can say that “x is congruent to y modulor z.”

There is a property of this relation which is very useful.

Suppose x1=y1(mod z) and x2=y2(mod z), then- x1.x2=y1.y2(mod z)

Now, coming to the original problem, suppose you have been given the seemingly tedious task of finding the remainder when a very large number x^r is divided by z. Let us tackle this problem with an example.

Suppose we want to find 3^48 divided by 11. Do we go about computing 3^48 and then dividing it by 11? Seemingly impossible! So what we do is we find ‘a’ in (3)^(2k) = a mod 11 for every 2k less than or equal to r(=48).

Hence, 3^2 = 9 (mod 11)
3^4 = 81 = 4 (mod 11)
3^8 = 3^4.3^4 = 4^2 (mod 11) = 5 (mod 11) (Realize how easy finding 3^8 mod 11 was using the property stated above!)
Similarly,
3^16 = 5^2 (mod 11) = 3 (mod 11)
3^32 = 3^2 (mod 11) = 9 (mod 11)

Now, note that 48 = 2^6 + 2^5 = 32 + 16.

So to find 3^48=a mod 11, we write a (mod 11) = 6*3 (mod 11). (Again using the property)

3^48 = 3^(16+32)
= 3^16.3^32 = 3*9 (mod 11) = 5

Hence, we can very easily find the remainder, on paper, without using supercomputers, when a large number is divided by some smaller number.

  1. General division rule in Vedic Mathematics.

 

This might seem to you a little complicated in the beginning. But a little careful analysis will find your doubts melting. Mastery over this method once done will replace the conventional methodology of dividing huge numbers wherein decimal digits are required rather than just finding the remainder.

 

This method will be explained by means of examples;

 

  1. Divide 716769 by 54;

 

Reduce the divisor 54 to 5 pushing the remaining digit 4 “on top of the flag”. Corresponding to the number of digits flagged on top (in this case, one), the rightmost part of the number to be divided is split to mark the placeholder of the decimal point or the remainder portion.

The following are the steps to be done;

 

  1. 7 ÷ 5 = 1 remainder 2. Put the quotient 1, the first digit of the solution, in the first box of the bottom row and carry over the remainder 2
  2. The product of the flagged number (4) and the previous quotient (1) must be subtracted from the next number (21) before the division can proceed. 21 – 4 x 1 = 1717 ÷ 5 = 3 remainder 2. Put down the 3 and carry over the 2
  3. Again subtract the product of the flagged number (4) and the previous quotient (3), 26 – 4 x 3 = 1414 ÷ 5 = 2 remainder 4. Put down the 2 and carry over the 4
  4. 47 – 4 x 2 = 3939 ÷ 5 = 7 remainder 4. Put down the 7 and carry over the 4
  5. 46 – 4 x 7 = 1818 ÷ 5 = 3 remainder 3. Put down the 3 and carry over the 3
  6. 39 – 4 x 3 = 27. Since the decimal point is reached here, 27 is the raw remainder. If decimal places are required, the division can proceed as before, filling the original number with zeros after the decimal point27 ÷ 5 = 5 remainder 2. Put down the 5 (after the decimal point) and carry over the 2
  7. 20 – 4 x 5 = 0. There is nothing left to divide, so this cleanly completes the division

 

Thus the answer is 13273.5 as reflected in the second box.

 

  1. Divide 45026 by 47;

Reduce the divisor 47 to 4 pushing the remaining digit 7 “on top of the flag”. Corresponding to the number of digits flagged on top (in this case, one), the rightmost part of the number to be divided is split to mark the placeholder of the decimal point or the remainder portion.

                                                             4                      9                      7                       5

4, (7)

4

5

0

2

6

 

0

9

5

8

0

 

  1. 4 ÷ 4 = 0 remainder 4. Put the quotient 0, the first digit of the solution, in the first box of the bottom row and carry over the remainder 4
  2. The product of the flagged number (7) and the previous quotient (0) must be subtracted from the next number (45) before the division can proceed. 45 – 7 x 0 = 4545 ÷ 4 = 9 remainder 9. Put down the quotient 9 and carry over the remainder 9.
  3. Again subtract the product of the flagged number (7) and the previous quotient (9), 90 – 7 x 9 = 2727 ÷ 4 = 5 remainder 7. Put down the quotient 5 and carry over the remainder 7.
  4. 72 – 7 x 5 = 3737 ÷ 4 = 8 remainder 5. Put down the quotient 8 and carry over the remainder 5.
  5. 56 – 7 x 8 = 0there is nothing left to divide, so this cleanly completes the division.

 

Thus the answer is 958.0


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Basic Division

Division is the process of splitting a number into equal parts. The number being divided is called the dividend, the number doing the dividing is called the divisor, and the result is called the quotient.

There are many different ways to divide numbers. The most common way is to use long division. Long division is a step-by-step process that can be used to divide any two numbers.

To divide a number by a single digit, you can use the shortcut method. The shortcut method is a quick way to divide a number by a single digit.

To divide a number by a two-digit number, you can use the box method. The box method is a visual way to divide a number by a two-digit number.

FRACTIONS

A fraction is a part of a whole. It is written as two numbers, one on top of the other, with a line between them. The number on top is called the numerator, and the number on the bottom is called the denominator.

To add or subtract fractions, you need to make sure that the denominators are the same. Then, you can add or subtract the numerators.

To multiply fractions, you multiply the numerators and the denominators.

To divide fractions, you flip the second fraction upside down and multiply.

Decimals

A decimal is a number that is written with a decimal point. The decimal point separates the whole part of the number from the fractional part of the number.

To add or subtract decimals, you need to make sure that the decimal points are lined up. Then, you can add or subtract the numbers as usual.

To multiply decimals, you multiply the whole numbers and the decimals separately. Then, you add the decimal points in the product.

To divide decimals, you move the decimal point in the divisor to the right the same number of places as there are digits after the decimal point in the dividend. Then, you divide as usual.

Percents

A percent is a number that is written out of one hundred. It is written with a percent sign (%).

To convert a decimal to a percent, you multiply the decimal by 100 and add a percent sign.

To convert a percent to a decimal, you divide the percent by 100.

Mental Math

Mental math is the ability to do math problems in your head. There are many different ways to do mental math. One way is to use shortcuts. Shortcuts are quick ways to solve math problems.

Another way to do mental math is to use patterns. Patterns are repeated sequences of numbers. You can use patterns to help you solve math problems.

Estimation

Estimation is the ability to guess the answer to a math problem without actually solving it. There are many different ways to estimate. One way is to round the numbers in the problem.

Another way to estimate is to use benchmarks. Benchmarks are known values that you can use to compare other numbers.

Word Problems

A word problem is a math problem that is written in words. To solve a word problem, you need to read the problem carefully and identify the key information. Then, you can use that information to solve the problem.

There are many different types of word problems. Some common types of word problems include:

Word problems can be challenging, but they can also be very rewarding. By following the steps above, you can learn to solve word problems quickly and accurately.

Division

Division is the process of splitting a number into equal parts. The number being divided is called the dividend, the number of parts is called the divisor, and the result is called the quotient.

Frequently Asked Questions

1. What is the shortcut for division?

The shortcut for division is multiplication. To divide a number by another number, you can multiply the number by the reciprocal of the divisor. The reciprocal of a number is the number that, when multiplied by the original number, equals 1.

For example, to divide 12 by 3, you can multiply 12 by 1/3. 1/3 is the reciprocal of 3, so 12 x 1/3 = 4.

2. What are some tips for doing division quickly?

Here are some tips for doing division quickly:

3. What are some common mistakes people make when dividing?

Here are some common mistakes people make when dividing:

4. How can I improve my division skills?

Here are some tips for improving your division skills:

  1. What is the fastest way to divide 123 by 4?
    (A) 30 + 12 + 3 = 45
    (B) 30 + 6 = 36
    (C) 30 + 3 = 33
    (D) 30 + 1 = 31

  2. What is the fastest way to divide 147 by 7?
    (A) 21 + 2 + 4 = 27
    (B) 21 + 2 = 23
    (C) 21 + 1 = 22
    (D) 21 + 0 = 21

  3. What is the fastest way to divide 198 by 9?
    (A) 22 + 2 + 4 = 28
    (B) 22 + 2 = 24
    (C) 22 + 1 = 23
    (D) 22 + 0 = 22

  4. What is the fastest way to divide 245 by 11?
    (A) 22 + 2 + 1 = 25
    (B) 22 + 2 = 24
    (C) 22 + 1 = 23
    (D) 22 + 0 = 22

  5. What is the fastest way to divide 312 by 13?
    (A) 24 + 2 + 2 = 28
    (B) 24 + 2 = 26
    (C) 24 + 1 = 25
    (D) 24 + 0 = 24

  6. What is the fastest way to divide 378 by 17?
    (A) 22 + 2 + 2 + 2 = 28
    (B) 22 + 2 + 2 = 26
    (C) 22 + 2 + 1 = 25
    (D) 22 + 2 + 0 = 24

  7. What is the fastest way to divide 435 by 19?
    (A) 23 + 2 + 2 + 2 = 29
    (B) 23 + 2 + 2 = 27
    (C) 23 + 2 + 1 = 26
    (D) 23 + 2 + 0 = 24

  8. What is the fastest way to divide 502 by 23?
    (A) 22 + 2 + 2 + 2 + 2 = 30
    (B) 22 + 2 + 2 + 2 = 28
    (C) 22 + 2 + 2 + 1 = 27
    (D) 22 + 2 + 2 + 0 = 26

  9. What is the fastest way to divide 569 by 29?
    (A) 20 + 2 + 2 + 2 + 2 + 2 = 30
    (B) 20 + 2 + 2 + 2 + 2 = 28
    (C) 20 + 2 + 2 + 2 + 1 = 27
    (D) 20 + 2 + 2 + 2 + 0 = 26

  10. What is the fastest way to divide 636 by 31?
    (A) 20 + 2 + 2 + 2 + 2 + 2 + 2 = 32
    (B) 20 + 2 + 2 + 2 + 2 + 2 = 28
    (C) 20 + 2 + 2 + 2 + 2 + 1 = 27
    (D) 20 + 2 + 2 + 2 + 2 + 0 = 26

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