SWEET MATHS Prepared by O.MADHAVARAJ

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Mar 18, 20150 comments


  


Dedicated to my parents

Prepared by O. MADHAVARAJ


                                               SWEET MATHS

For most of the students, the subject of Mathematics is a bitter lemon, to make it as sweet lemon , the following some useful tricks on mathematical problems I have seen so far from the books of “Secret of mental maths “ and “Marvelous Maths tricks “ and the oldest script of “Vedic Maths “ were given below for the use of students.
                   

The secret of success in the mathematics is depends on the quickest way to get the answers for the problems.


The object of this essay is to enlighten the easy way and tricks in the mathematics problem.


                                                  ADDITION
                             Adding consecutive series

1.Eg: 1,2,3,4,5,6,7,8,9

S.C : Multiply last number with  one more than that number and divide by 2 ie nx(n+1)/2

Multiply 9x9+1=90
Divide    90/2 =45

Result  =45


2.Eg : 33,34,35,36,37,38,39,40,41

S.C :  Add the lowest number with the highest, multilply with total number in group and then divide by 2.

Add  33+41=74
Multiply 74x9 =666
Divide  666/2 =333

Result =333
  


3.Find the sum of all odd numbers in between a series

Eg:  1,3,5,7,9,11 …………99

S.C : Square the number in the series

Here the total number in series is 50

Square 50^2     =2500

Result =2500

4.Find the sum of all even numbers in a series

Eg: 2,4,6,8,10 ………..100

S.C : Multiply total numbers in series by more one

Here the total number in series =50

Multilply   50x50+1=2550

Result =2550


5.Adding a series having a common difference

Eg:  44,47,50,53

S.C: Add the lowest with highest number, divide b 2 then multiply with the total number in series.

Add   44+53 =97
Divide 97/2
Multiply 97/2x4 =194

Result =194








6.Adding a series having a common ratio

Eg:  44,88,176,352

S.C: Multiply the common ratio by itself as many times as the series having number, subtract one and multiply with first number in series.Divide result by less than one ratio  

Multiply the ratio 4 times 2x2x2x2 =16
Less one 16-1 =15
Multiply  15x44=660
Divide by 2-1 =1 ie 660/1=660

Result =660

7.Adding a sequence in the form 13+23+33…..103

S.C:  Square the sum of the series

Sum of the series =10x11/2=55
Square                  == 3025


8.Adding infinite series in the form a+a/b+a/b2 ……  

S.C: the first term divided by one subtract by the common term multiplied in series

Eg: 6+4+8/3…..

The common factor multiplied is   2/3

6 divided 1/3 =6x3/1=18

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                                                         MULTIPLICATION

9.Multiplying by 1001:

27848x1001

S.C: Write first 3 digits as it is ,for fourth  add first digit with 4th digit and for 5th digit add second digit with last digit , then the last 3 digits as it is given.

First 3 digit =278(multilier 4 digit hence first 3 digit as in the number)
Add 2+4 =6
Add 7+8 =15(if more than 10 the remaining to be added with previous digit)
Last three =848

Now  answer  278615848 =27875848

Result 27875848

10.Multiplying by 100001:

423456x100001

S.C: Write first 5 digits as it is ,for sixth  add first digit with last digit , then the last 5 digits as it is given.

First 5 digit =42345(multilier 6 digit hence first 5 digit as in the number)
Add 6+4 =10
Last 5 digits =23456

Now  answer  423451023456 =42346023456

Result 42346023456

11. Eg: 234567x1000001

Multiplier is 7 digit hence the first six digit of answer is 234567
As there is no 7th digit to add the last digits were 234567

Hence answer 234567234567







12.Multiplying 2 digit number both end in 5 and one starting with odd number

S.C:To the product of the ten digits add one half of their sum(ignore fraction) affix 75 to the result

Eg: 95x45

The product of ten digit 9x4 =36
Add one half of their sum =1/2(9+4) = 6 (ignore fraction) =36+6=42

Result 4275

13.Multiplying 2 digit number both end in 5 and both starting with even number

S.C:To the product of the ten digits add one half of their sum(ignore fraction) affix 25 to the result

Eg: 85x45

The product of ten digit 8x4 =32
Add one half of their sum =1/2(8+4) = 6 (ignore fraction) =32+6=38

Result 3825

14.Multiling two two digit numbers whose tens digits are both 5 and other digit both odd or both even

Eg:52x58

S.C: Add one-half the sum of the tens digits to 25.Affix the product of the last digits to the result. If the product is less than 10 proceeds with 0

Add one half of he sum of digits 5+5 =10  =5+25=30
Product of the units =2x8 =16

Result =3016










15.Multipling two  digit numbers whose tens digits are both 5 and other digit one odd and other even

Eg:52x59

S.C: Add one-half the sum of the tens digits to 25.Affix 50 to the product of the last digits to the result.

Add one half of he sum of digits 5+5 =10  =5+25=30
Product of the units =2x9+50 =68

Result =3068

Now from right to left

16.Multilying by 11

Eg: 3421634x11

S.C: Add the neighbor to the number


Last  4 +neighbor 0             =4
Next 3+neighbor 4               =7
Next 6+neighbor 3               =9
Next 1+neighbor 6               =7
Next 2+neighbor 1               =3
Next 4+neighbor 2               =6
Next 3+neighbor 4               =7
Next 0+neighbor 3              =3

Result  = 37637974

17.Eg: 377x11

Last  7 +neighbor 0             =7
Next 7+neighbor 7               =14
Next 3+neighbor 7               =10+1=11
Next 0+neighbor 3               =3+1

Result =4147



18.Multiplying by 12

Eg:3421634x12

S.C:Double the number add the neighbor

Double the Last  4 +neighbor 0             =8
Next double the  3+neighbor 4               =10
Next double the  6+neighbor 3               =15+1
Next double the  1+neighbor 6               =8+1
Next double  the 2+neighbor 1               =5
Next double the  4+neighbor 2               =10
Next double the  3+neighbor 4               =10+1
Next double the  0+neighbor 3              =3+1

Result  = 41059608

19.Squaring any number ending with 5

Eg:  1952

S.C : Multiply the complete number left to 5 with one more and affix 25 to the result

The number left to 5 is 19
Multiply with one more ie 19x(19+1) =380
Affix 25 ie 38025

Result  =38025

20.Squaring any 3 digit number ending in 25

Eg: 6252

S.C: Add one half of hundredth digit with square of the hundredth digit (if it comes single digit then the result is 10 thousand digit else 100 thousand and 10 thousand digits),if the hundredth digit is odd number then 5 else 0 , suffix 625

First, square the hundredth digit ie 62 =36
Add one half of the hundredth digit ie 6/2=3+36=39(100 thousand and 10 thousand digits)
The thousand digit number is zero (hundredth digit even )
Suffix 625  with above 390625

Result =390625

21.Eg: 5252

 First, square the hundredth digit ie 52 =25
Add one half of the hundredth digit ie 5/2=2+25=27(100 thousand and 10 thousand digits)
The thousand digit number is 5 (hundredth digit odd )
Suffix 625  with above 275625

Result =275625

22.Squaring any number ending in 9

Eg: 1492

S.C: Multiply the to the left of 9 by two more than itself,substract twice the number to the left of 9,affix 1 to the result

Multiply the number to the left of 9 with two more than that number =14x14+2=224
Affix 8  =2248
Substract twice the number left side of 9 ie2248-( 2x14)=2220
Affix 1 to the above ie 22201

Result =22201

23.Squaring any number having 9 only

Eg: 9992

S.C: Write one less 9 in the given number followed b 8 and then followed by  as many zeros as nine with suffix 1

One 9 less =99
Followed b 8 =998
Zeros         99800
Suffix 1     998001

Result  998001

24.The trick with 37

Eg: 37x3  =111 say 37x3n =nnn

Another Eg: 37x15 =37x(3x5)=555



25.The trick with 25

25x4=100
25x35 = 35/4=8 reminder 3x25 =875

26.Multiply by 99,999,9999 etc

56x99  

56-1 =55
100-56 =44
Result =5544

567x999

567-1=566
1000-567=433
Result =566433

10245x99999
10245-1=10244
100000-10245=89755
Result =1024489755
                                     
27.Cross multiplication

Eg :  35x35

3            5
3            5
a) first multiply RHS 5x5 =25                                              5 remaining(2)
b) second multiply cross 3x5+3x5 =30     30+2                   2                  (3)    
c) thirdly multiply LHS 3x3 =9                  9+3                  12

Result :   1225    
                         
Splitting into two as one with multiple of 10 

Eg: 962

Add the numbers 96+96 = 192
Split into two like 90,102
Multiply both splited 90x102 =9180
Add 6x6                                    =9180+63 =9216 

Result =9216
28.Base method: base 10
Eg : 328x377

328                                    +28
      377                  +77

300                                    28x77 =2156
      105 (28+77)
     -------
      405 x30  =12150           (30x10)
                           2156
                       ----------
                       123656
                       ----------

Result =123656

29.Eg:   112x87 
              
             112              +12
               87               -13

100                              + 12x-13 = -156 
  97 (12-13) & -2 for substracting 156
------
97x10  =970                        (10x10)
                  44(200-156)
                --------    
                 9744  
                             --------
Result  =9744
 
30.Eg:   87x88        

87                                                -13
88                                                -12

100                                              -12x-13 =156
          75(100-25) ie -13-12

          75x10    =750
                             156
                           ---------   
                           7656
  
Result  =7656       

31.Gelosia method:

Eg: 89x67


8                   9                                                                            
4                     8
5    
          6
  5    
          4
6    
          3


First step 8x6 =48    4 in upper portion and 8 in lower portion
Second    8x7 =56    5                            and 6                          
Third       9x6  =54   5                            and 4 and so on
               3      = 3
4+6+6=18       =16
5+8+5=18       = 19
               4      =5
Result =5963
           

32.WEIRD MULTIPLICATION:

Eg : 34x23

  3                       4
 
                                          2
 

                                          3
 



The intersection points 6                     =7
The intersection points 17                   =18
The intersection points 12                    =12


Result     =782  



33.Russian Multiplication
  
Eg: 36X23

First half the LHS and Double RHS (ignore fraction) till LHS become 1
Say   18        46 
9            92
     4      184
     2      368
           1      736   
 Add RHS when LHS odd number ie 92+736=828 

Result    =828

34.Paper Strip Multiplication :

Eg:    56x43

First reverse the multiplier
Write the numbers in paper strip
And kept as below and multily



    6  5
 

    4    3
 
 




6x3  =1

    6  5
 
 


    4  3
 
    


   6x4 +5x3 =39+1 (reminder) =40   

    6  5
 
 
                       

    4  3
 
 


5x4  =20+4=24

Result =2408



35.DUPLEX METHOD:

Eg: 45x57

Add zero before the numbers ie 045x057

Step 1  5x7     =35 (single multiplication)
Step 2  5x5+4x7=56 (two multiplication)
Step 3  5x0+4x5+0x7=20(three multiplication)

Result  =        2565

36.Eg: 35x24

Make it as a+b,c+d
(a+b)x(c+d) = (ac+ad)+(bc+bd)
Say   (30+5)x(20+4) = 600+120+100+20 =840
Result =840

37.Eg:35x24

Make it as a-b,c-d
(a-b)x(c-d) = (ac-ad)-(bc-bd)
Say   (40-5)x(30-6) = 1200-240-150+30 =840
Result =840


Squaring


38.Eg: 352

Surplus to base 10      3x10+5  = 5
Number+surplus     35+5         =40
Surplus square                         =25

LHS 40x3                                 =120
RHS 25                                     =  25

                                          =       120 25  =1225






39.Eg: 352

deficit to base 10      4x10-5  = 5
Number-deficit         35-5      =30

deficit square                         =25

LHS 30x4                                 =120
RHS 25                                     =  25

                                          =       120 25  =1225

40.Eg:   752


First step 52                 =25
Second step 2x7x5      =70
Third step 72                    =49

Answer is         56 72 25 =5625

THREE DIGITS MULTIPLICATION

41.Eg: 123x234

a) first multiply RHS 3x4 =12                                                    2 remaining(1)
b) second multiply cross 2x4+3x3 =17            17+1                  8                  (1)
c) thirdly cross with straight 1x4+2x3+2x3      16+1                 7                 (1)
d) second cross 1x3+2x2                                   7+1                   8
e) final multiply LHS 1x2                                                          2

Result  28782

42.Alternate 12 3x 23 4


a) first multiply RHS 3x4=12                                              2 remaining(1)
b) second multiply cross 12x4+3x23 =117+1                     8                  (11)    
c) thirdly multiply LHS 12x23 =276+11                        287

Result                   28782





43.Eg: 123x117 Base 100

123                    +23
      117          +17
------------------------
140                  23x17  = +391

14000
    391
--------
14391
--------
44.Eg: 123x117  base 120

123              +3
117                -3 
------------------
120             -9  (123-3 or 117+3)
120x12 =14400
-          9
                ---------
                14391
                ---------

Gelosia method:

45.Eg: 123x234
   

0              0                0
    2                3                  4

0              0               0
       4                6             8
   0              0             1

          6           9                2
 
      1             2                 3


 







 2                                 =     2
9+1+8                          =   18
6+0+6+0+4=16+1       =   17            
0+4+0+3+0=7+1         =     8
0+2+0                          =     2
0                                  =     0
Result   =28782
Duplex  method:

46. Eg: 123x456

Add two zero on both number ie 00123x00456

a) single digit 3x6                                                   =18
b) two digit 23x56 =2x6+3x5=27+1                      =28
c) three digit 123x456 =3x4+2x5+1x6 =28           =30
d)four digit 0123x0456 =0x6+1x5+2x4+3x0=13  =16
e) Five digit 00123x00456=4+1                             =5

Result  =56088


                                         Cube

47.Eg:  124 3


12    4
        a  b           b/a=4/12=1/3

a 3    =12 =1728

1728    1728x1/3     (1728x1/3)x1/3    [(1728x1/3)x1/3]x13 (a3  a2b  abb3 )

1728       576                 192                      64
             1152                 384                                    (1   2   2   1)

--------------------------------------------------------   
1728    1728                 576                      64

                                         64     = 4 (6)
                                  6+576      = 2 (58)
                              58+1728     =  6 (178)
                            1728+178       =  1906
Result =1906624



48.Eg: 233

A3=(A-D)(A+D)A+D2A here A=23 D=3

    = 20x26x23+32*23 =12167

49.Eg: 63 4

6        3
       a    b       b/a= 3/6=1/2                    (a4  a3b  a2b2  ab3  b4)

a4= 64=1296

1296          648          324           162           81
                1944        1620           486                           (1  3    5   3   1 )
--------------------------------------------------------
1296       2592         1944           648           81
--------------------------------------------------------


81      =  1 (8)
                       648+8   = 6 (65)
                     1944+65 = 9(200)
                     2592+200=2(279)
                   1296+279 = 1575

Result   15752961



50.Eg: 1543

15      4
       a   b      

a3 =153 =3375

3375     3375x4/15   (3375x4/15)x4/15     [(3375x4/15)x4/15]x4/15

3375               900                 240                          64 
                    1800                  480                                 
----------------------------------------------------------------- 
3375           2700                    720                        64

                            64        =   4  (6)
                       6+720      =    6  (72)
                    72+2700     =    2  (277)
                  277+3375    =     3652

Result =3652264

                                            SQUARE ROOT

51.Eg: √169

From right mark two two digit
First digit is 1   hence the square root having 1 as first digit
Last digit =9 hence the digit ends with either 3 or 7 (square 9 or 49)
Sum of the digits =1+6+9 =16=7

Either 13 or 17  1+3=7   or 1+49=5

Hence answer is 13

52.Eg : √9216

92      16
First two digit is 92 hence first digit be 9
The last digit is 16 ends in 6 hence either 4 or 6
Sum of the numbers 9+2+1+6 = 18=9
Answer should be either 94 or 96
9+4 =13=42 =16=7 not equal to 9
9+6 =15=62 =36=9
Result is 96

53.Eg: √15129

         1  51   29
First digit be 1
Last digit be 3 or 7
Sum of the digit =1+5+1+2+9 =18=9
151 is between 122  hence the middle is 2
123 or 127
1+2+3 =6 or 1+2+7 =10=1 squaring 6 =9 hence last digit is 3

Result 123

54.Bakshali formula :

Square root = a+b/2a

√76     

8   12
      a    b

8+12/16  = 8.75  but actual 8.71


Cube Root :

55.Eg : 3√ 1953125

    1  2   5
| 1        953  125                300(1)2 +30(1)(2) +22     =364
| 1
--------                                300(12)2+30(12)(5)+52 =45025
364|      953
             728
             -----
45025| 225125
225125
----------
      0
----------

56.Eg: √1953125

1 953 125

First digit be 1 
Last digit be 5
Sum of the numbers =1+9+5+3+1+2+5=26=8 hence middle digit 2

Result =125


DIVISION

57. A number divisible by 2

S.C: If the last digit is an even number

Eg : 41256 is divisible by 2

Last digit is 6 which is divisible by 2

Hence the number is divisible.




   
58.  A number divisible by 3

S.C: If the sum of the digits  is divisible by 3 then divisible

Eg : 41256 is divisible by 3

Sum of the digits 4+1+2+5+6 =18 which is divisible by 3 

Hence the number is divisible.

59. A number divisible by 4

S.C: If the last two digit is an divisible by 4

Eg : 41256 is divisible by 4

Last two digit is 56 which is divisible by 4

Hence the number is divisible.



60. A number divisible by 5

S.C: If the last digit is either 0 or 5 is divisible

Eg : 41256 is divisible by 5

Last digit is 6 
Hence the number is not divisible.


                 
61. A number divisible by 6

S.C: If the last digit is an even number and the sum of the number is divisible by 3
Eg : 41256 is divisible by 6

Last digit is 6 which is divisible by 2
Sum of the digits 4+1+2+5+6 = 18 is divisible by 3
Hence the number is divisible.





62.A number divisible by 8

S.C: If the last three digits is  divisible by 8

Eg : 41256 is divisible by 8

Last two digit is 256 which is divisible by 8

Hence the number is divisible.

63.Eg: 8678991 divisible by 19

Osculator =2

2x1+9 =             11
(2x1)+1+9 =      12
(2x2)+1+8 =      13
(2x3)+1+7 =      14
(2x4)+1+6 =      15
(2x5)+1+8 =      19

19/19 hence it is divisible

Number         +osculator          -osculator       total

    1                     1                           0                 1
    3                     1                           2                 3
    7                     5                           2                 7
    9                     1                           8                 9
   11                   10                          1                11
   13                     4                          9                13
   17                    12                         5                17


64.Eg: 17160384 divisible by 139

14x4+8                            =64
14x4+6+3                        =65
14x5+6+0                        =76                                   
14x6+7+6                        =97
14x7+9+1                        =108
14x8+10+7                      =129
14x9+12+1                      =139

Divisible by 139

65. Eg: 3245693 divisible by 11

Odd 3+4+ 6+3 = 16
Even 2+5+9     =16  both are equal hence divisible.


66.Eg:132101 / 9
 
132101         13210   1                        1          1
                       1467    7                     1+3         4
                      14677   8                   1+3+2       6
                                                      1+3+2+1     7
                                                    1+3+2+1+0   7
         14677 reminder 8                                       

67.Eg: 23243 /9

2324   3                                2               2
 257   11                           2+3               5  
         -----                       2+2+3             7
2581   14                      2+2+3+4          11    
2582 reminder     5

68. Eg : 2679502 / 43
       6 2 3 1 4
     -----------------------
43|26 7 9 5 0 2
     24
--------
        2 7-(3x6) =9
                          8
                     ------ 
                          19 –(3x2) = 13
                                              12
                                           -------
                                                 1 5 –(3x3) =6
                                                                     4
                                                                 -----
                                                                     2 0 -3x1=17
                                                                                    16
                                                                                  -----
                                                                                       12 -12 =0


Result    62314

69.Eg: 123456/69

     1 7 8 9
  ---------------------
7|12  3  4  5  6               more than one ie 69+1=70
     7
   ----
     5 3+1 =54
                  49 
                ----- 
                     54+7=61
                               56
                              ---- 
                                 5 5+8 =63
                                              63
                                              ---
                                               0+9 +6 =15

Result   1789 reminder 15




70.Eg: 738704 divide by 79

       9 3 5 0
   -------------------
8| 73   8  7  0   4
    72
   -----
       18+9 =27
                  24
                 -----
                    3 7+3 =40
                                 40
                              ------ 
                                  54 
                              ------

Result 9350 reminder 54





71.Eg: 113989/21

     5 4 2 8
  --------------
2| 11  3  9  8   9
    10
 ------
      13-5 =8 
                 8
           ------- 
                 9 -4 =5
                          4
                     ------
                           18-2=16
                                     16
                                   ----- 
                                       09-8 =1

Result 5428 reminder 1

72.Eg: 2042

LHS 22 (2x4x2) 42 RHS

Result = 41616

73.Eg:3082

LHS32(3x8x2)82RHS

Result = 94864















74.Find the value of 1/19
      0.52631578947368
   --------------------------------
2| 10
    10
------ 
     5
     4
------
     10+2=12
               12
             -----
                 6
                 6
               ----
                 3
                 2
               ----
                 10+1=11
                           10
                        ------
                             10+5 =15
                                        14
                                      -----
                                          10+7=17
                                                    16
                                                    ----
                                                       18
                                                       18
                                                     -----
                                                         9
                                                         8
                                                     -----
                                                         10+4=14
                                                                    14
                                                                   -----
                                                                      7
                                                                      6
                                                                  ------
                                                                      10+3 =13
                                                                                 12
                                                                              -----
                                                                                  10+6=16
                                                                                             16
                                                                                         ------  
Result  =0.52631578947368

75. Find the value of 1/19

For a fraction of the form in whose denominator 9 is the last digit, we take the case of 1 / 19 as follows:
For 1 / 19, 'previous' of 19 is 1. And one more than of it is 1 + 1 = 2.
Therefore 2 is the multiplier for the conversion. We write the last digit in the numerator as 1 and follow the steps leftwards.

 Step. 2 : 21(multiply 1 by 2, put to left)
Step. 3 : 421(multiply 2 by 2, put to left)
Step. 4 : 8421(multiply 4 by 2, put to left) and so on

                                                  1
                                                21 
                                              421
                                            8421
                                         168421
                                       1368421
                                       7368421
                                    147368421
                                    947368421
                                 18947368421
                               178947368421
                             1578947368421
                           11578947368421
                           31578947368421
                         631578947368421
                      12631578947368421
                     52631578947368421
                0. 052631578947368421




76.Eg: 113989 divided by 113

1 1 3    11  3  9  8 9
  -1-3    -1 -3
                  0  0
                     -0 -0
                           -9 -27
----------------------------------
              1009     -1 -18        1008 R 113-28=85  



77.Eg: 124992 divided by 124

1 2 4       1  2  4  9  9     2
  -2-4         -2 -4
                       0  0
                           0   0
                               -18 -36
----------------------------------- 
                 1 0  0  9  -9 -34
 
Q 1009 R-(34+90) =1 ie 1008


78.Eg: 1227352 divided by 9898

Divisor nearer to 10000 here deficit =10000-9898= 0102
Keep last four digits for reminder

0102   | 1 2  2    7 3 5 2
               0   1    0 2
                    0    2 0 4
                          0 3 0  6
----------------------------- 
            1 2  3     9898
            123+1 =124
Q=124
                                

79.Eg: 111500 by 892

108|1 1 1  5 0 0
          1 0 8
             2 0 16
                3  0  24
----------------------
123      1784 =123+2=125


80.Eg: 97092 by 783

217| 9 7  0 9 2
          19 5 3                 217x9
              55 4 2             217x26
-------------------
       9 26 60 16 4 =4+16+6000=6164=783x8
90+26+8 =124
81. Karatsuba method of multiplication  :two digits

Eg: 78x21  

axb= u x102+(u+w-v)x10+w

Here u=7x2 =14
v=(8-7)x(1-2) = -1
w=8x1 =8
1400+230+8 =1638

82.Eg: 87x34

u= 8x3 =24
v =(8-7)x(4-3)=1
w=7x4=28

2400+530+28=2958

83.Karatsuba method of multiplication  :four digits

axb = ux104+(u+w-v)x102+w

Eg: 5678x4321

u=56x43=2408
v=78-56x21-43=22x-22= -484
w=78x21=1638

24080000+453000+1638 =24534638


84.Fourier Techniques

Eg:123x654

p(x) =3x0+2x+1x2
q(x) =4x0+5x+6x2

      =3.4(x0)+x(3.5+4.2)+ x2(3.6+4.1+2.5)+ x3(2.6+5.1)+1.6x4   
      = 12+230+3200+17000+60000=80442






85.Eg: 54612 divided by 246

          222
        ----------
24 6|54 6 1 2
        48
      ------
           66-12=54
                       48
                    ------
                         6 1-12=49
                                     48
                                    -----
                                        12-12 =0
Q=222

86.Eg: 28949025 divided by 2345
           12345
          ----------------
234 5|289 4 9 0 2 5
         234
         ------
          55 4-5=549
                      468
                     ------
                        819 -10 =809
                                        702
                                      -------
                                       1070 -15 =1055
                                                          936
                                                        ------
                                                          1192-20=1172
                                                                         1170
                                                                        -------
                                                                              25-25 =0 
Q=12345











87.  14121 divided by 99

    99   |  1  41  21
              1 | 1+41|41+21
              1     42      62

          Result Q= 142  R62

    Another one  Eg: 41089 divided by 33

     33| 4 10 89
           4  4+10 4+10+89
           414       103
           414      103/33=3 R4
           414x3+3 =1242+3 =1245 R4
           Q=1245 R 4 

88.  Partial Quatients Method:

                1220 divide by 16

16| 1220
        800    50
       -------
         420
          320    20
        ------ 
          100
            80       5
          ----- 
            20   
            16        1
           -----
               4

Result 50+20+5+1 =76 R 4










Some algebra problem

89.Eg: (a+2b)(3a+b)

Cross multiplication method:

a  +  2b
3a +   b

2bxb     =2b2
ab+6ab  =7ab
ax3a      =3a2

Result =3a2+7ab+2b2

90. Eg: (4x2+3)(5x+6)

4x2 +0x+3
0x2+5x+6

3x6                                                             =18
(5x)(3)+(0x)(6)                                          =15x
(4x2)(6)+(0x2)(3)+(0x)(5x)                        =24x2
(4x2)(5x) +(ox2)(ox)                                  =20x3
(4x2)(0x2)                                                 =0

Result              20x3+24x2+15x+18

91. x3+5x2+3x+7 divided by x-2
         x+7x +17
      -----------------------
x-2 | x3 + 5x2 + 3x +7
        x3
       ----  
        5x2 +2x2 =7x2
                         7x2
                       -----   
                          3x +14x=17x
                                         17x
                                       ------
                                         7+34=41
Result   =x2+7x+17  R 41




92. Eg :x3-3x2+10x-4  by x-5
         1 +2 +20
     ----------------
x-5| 1  -3   +10   - 4
          +5
         ---------
                   10
                  ----  
                          100
----------------------------------
      1 +2+20    96

Result = x2+2x+20 R 9

93. Four digit multiplication:

Eg: 1188x1212

Cross multiplication

1. 8x2                                                            =16
2. 8x2+1x8                                                    =24+1 =25
3.1x2+2x8+1x8                                             =26+2=28
4.1x2+1x8+1x1+2x8                                    = 27+2=29
5.1x1+1x8+1x2                                            = 11+2=13
6.1x2+1x1                                                    =3+1 =4
7.1x1                                                            =1

Result  =1439856

94. Squaring a four digit number

Eg:12342

Surplus 1234-1200=34 base 1200
Surplus+number =1234+34=1268
Surplus square  =34 =1156

1268 |1156
x12
15216
      1156
Result =1522756


                                               Magic number
1/9 =0.111111
2/9 =0.222222
3/9=0.3333333
4/9=0.4444444
5/9=0.5555555
6/9=0.6666666
7/9=0.7777777
8/9=0.8888888

95.Eg: 600/9 =0.66666x100=66.66

1x1                                 =           1
11x11                             =         121
111x111                         =       12321
1111x1111                     =     1234321
11111x11111                 =    123454321
111111x111111             =   12345654321
1111111x1111111         = 1234567654321
11111111x11111111     =123456787654321


12345679x9           =111111111
12345679x18         =222222222
12345679x27         =333333333


135              =1+32+53
175              =1+72+53
518              =5+12+83

Fifth root :

96.Eg:8,58,73,40,257

x             x5
1            100 thousands
2               3 million
3             24 million
4            100 million
5            300 million
6            777 million
7              1.6  billion or 160 crores
8              3 billion  or 300 crores
9              6 billion or 600 crores



In the above example number is more than 600 crores hence the first digit =9
The last digit is the last digit of the number =7

Result =97

97.Eg: 39135393

Having more than 24 million hence first digit is 3
Last digit is                                                          3

Result  =33

98. Fourth root:

x             x4
 1            10000
 2            160000
 3            810000
 4           2560000
 5           6250000
 6         12960000
 7         24010000
 8         40960000
 9          65610000

99.Eg: 234256 

The first digit is  2
The last digit is either 2 or 4 or 6 or 8
Sum of the digits =2+3+4+2+5+6 =22 =4

22=2+2 =44 =256 =13=4
244 =64=1296 =18=9
264 =84 =4096 =19=1
284=104=1  
Hence 22 is the answer










100. Eg :37015056

The first digit is  7
The last digit is either 2 or 4 or 6 or 8
Sum of the digits =3+7+0+1+5+0+5+6 =27 =9

724=94=6561=18 =9
744=24=16=7
764=14 =1
784 =64=1296=18=9   

Here 704=24010000 and 72 is nearer to 70 and will have 2800000 but our number is 3700000 hence 78 is the correct answer.


Review : Multiplication

Eg: 123x234

Method 1:

1  2   3
2  3   4

a) 3x4                                         =12
b) 2x4+3x3  =17+1                    =18
c) 1x4+2x3+2x3 =16+1             =17          
d) 1x3+2x2  =7=1                      =8
e) 1x2                                         =2

Result        =28782

Method 2:

12            3
23            4

a) 3x4                                              =12
b)12x4+23x3  =117+1                    =118     
c) 23x12=276+11                           =287

Result   =28782 


Method 3:

123        -77
      234    +34      (base 200)
------------------- 
157        -77x34
157-15        (3000-2618)  15x200=3000
      142x2   =28400
                         382
                    --------
                    28782
                   ---------- 

Method 4:


0              0                0
    2                3                  4

0              0               0
       4                6             8
   0              0             1

          6           9                2
 
      1             2                 3


 







 2                                 =     2
9+1+8                          =   18
6+0+6+0+4=16+1       =   17            
0+4+0+3+0=7+1         =     8
0+2+0                          =     2
0                                  =     0
Result   =28782

Method 5:

123x234
00123x00234
a) single digit      3x4                                          =12    
b) double digit  23x34=3x3+2x4 =17+1             =18
c)  three digits  123x234=1x4+2x3+3x2  = 16+1 =17
d) four digits 0123x0234=0x4+1x3+2x2+3x0=7+1 =8
e)  five digits 00123x00234=0x4+0x3+1x2+2x0+3x0=2     
Result  =28782


Method 6 : 123x234

100x234 =23400
  20x234 =  4680
    3x234=     702
                  --------
                  28782
                ---------

Method 7:  

123           234
61           468
30           936
15        1872
  7    3754
  3    7518
          1  15036
             -----------
              28782    

Method 8:
1             2                     3

 










Point of intersection   =  12
                                    = 17
                                    = 16
                                    = 7
                                    =2
Result   =28782





Method 9 :

123x234

3     2        1
 

2     3        4
 
 






3x4               =12


3     2        1
 

2      3       4
 
 





3x3+2x4 =   17+1=18


3     2        1
 

2     3        4
 
 




3x2+2x3+1x4 = 16+1    =17


3     2        1
 

2     3        4
 
 





2x2+1x3           =7+1 =8

3     2        1
 
 



2       3     4
 
 



1x2                =2

Result      =28782



Method 10:

p(x) =3x0+2x+1x2
q(x) =4x0+3x+2x2

      =3.4(x0)+x(3.3+4.2)+ x2(1.4+2.3+2.3)+ x3(2.2+3.1)+1.2x4  
      = 12+170+1600+7000+20000=28782

Method 11:

u=12x23= 276
v=(3-12)(4-23) =171
w=3x4 =12

u100+(u+w-v)10+w
276x100+117x10=12=28782

Result = 28782

a)Eg: 2716032 is divisible by 22?
S.C : The last digit multiply by 1 and deduct from the remaining digits and so on till divisible by 22 or to become 0.
271603 2
        - 2
--------- 
27160 1
        1
--------
2715  9
      9
------
270   6
    6
-----
26    4
  4
----
22
22
----
0
---      


b) Eg:   3936 is divisible by 32

S.C : The last digit multiply by 3/2 and deduct from the remaining digits and so on till divisible by 32 or to become 0

393 6  
    9
-----
38  4
  6
----
32
32
----
0
---

c)Eg: 191814 is divisible by 42

S.C: The last digit multiply by 2 and deduct from the remaining digits and so on till divisible by 42 or to become 0

19181 4
        8
--------
1917  3
      6
------
191 1
    2
-------
18 9 
18
---
0
 Q= 9134/2=4567










d) Eg: 42036 is divisible by 62

S.C: The last digit multiply by 3 and deduct from the remaining digits and so on till divisible by 62 or to become 0

4203 6
    18
------
418 5
  15
----
40   3
  9
---
3   1
3
--
0

Q=1356/2=678

e)Eg: 1012290 is divisible by 82

S.C: The last digit multiply by 4 and deduct from the remaining digits and so on till divisible by 82 or to become 0

101229  0
        0
--------
10122   9
      36
-------
 1008     6   
     24
-----
     98     4    
     16
-------
     82
     82
-------
0        Q= 12345




f) Eg: 125868 is divisible by 102

S.C: The last digit multiply by 5 and deduct from the remaining digits and so on till divisible by 102 or to become 0

12586   8
      40
--------
1254      6
    30
------
122      4
  20
-----
102
102
-----
0
-----           Q=     1 468/2=234=1234


g) Eg: 315614 is divisible by 122

S.C: The last digit multiply by 6 and deduct from the remaining digits and so on till divisible by 122 or to become 0

31561   4
      24
--------
3153      7   
    42
------
311       1   
    6    
-----
30          5       
30
---
0                  5174/2=2587







i) Eg:14805 is divisible by 63

S.C: The last digit multiply by 2 and deduct from the remaining digits and so on till divisible by 63 or to  become 0

1480   5
    10
-------
147     0
    0
------
14       7
14 
----
0
---    
Q=705/3=235


j) Eg: 449934 is divisible by 123

S.C: The last digit multiply by 4 and deduct from the remaining digits and so on till divisible by 123 or to become 0

44993 4
      16
--------
4497   7
    28
------- 
446          9  
        36
--------
     41       0
       0
---------
4                  1
4
---
0                    Q =10974/3=3658
1                     





k) Eg:120717 divisible by 153

S.C: The last digit multiply by 5 and deduct from the remaining digits and so on till divisible by 153 or to become 0

12071   7  
      35   
-------- 
1203     6
    30
------
117       3
  15
----- 
 10       2
 10
----
0

Q=2367/3=789


l) Eg:8359074 is divisible by 183

S.C: The last digit multiply by  6 and deduct from the remaining digits and so on till divisible by 183 or to become 0

835907  4
        24
-----------
83588    3   
      18 
-------- 
8357       0  
      0
------ 
835        7
  42
---- 
79         3
18
--- 
6            1
6
--
0     Q =137034/3=45678

m) Eg: 434544 divisible by 44

S.C: The last digit multiply by 1 and deduct from the remaining digits and so on till divisible by 44 or to become 0
43454   4
        4
-------------
4345     0
      0
------  
434        5
    5
-----
42         9
  9
---
  3         3
  3
---
0             Q=39504/4=9876    

n) Eg:5313672 is divisible by 84

S.C: The last digit multiply by 2 and deduct from the remaining digits and so on till divisible by 84 or to become 0
531367  2
          4
---------  
53136    3 
        6
------------
5313      0
      0
------   
531        3  
    6        
------
   52        5
   10
-----
    4         2
    4
------
0              Q=253032/4=63258

o) Eg: 45756 is divisible by 124

S.C: The last digit multiply by 3 and deduct from the remaining digits and so on till divisible by 124 or to become 0

4575  6
    18
------ 
455    7  
  21 
-----  
43      4    
12
--- 
3         1
3    
----
0     
Q=1476/4=369


p) Eg : 5668988 is divisible by 164

S.C: The last digit multiply by 4 and deduct from the remaining digits and so on till divisible by 164 or to become 0

566898   8
        32
---------- 
56686     6
      24 
--------- 
5666        2
      8    
------- 
565          8
  32 
----- 
53            3
12   
--- 
4              1
4  
---
0            Q =138268/4 =34567 


Eg:   315 is divisible by 7

S.C : A number is of form 100a+b is divisible by 7 if and only 2a+b is divisible by 7 (babylonian method) 

3x100+15    a=3 b=15  
2a+b = 6+15=21 which is divisible by 7
Hence divisible

Eg: 168168 is divisible by 7

S.C: A six digit number of the form xyzxyz is divisible by 7


Eg: 1353 is divisible by 11

135| 3   Multily last digit by -1
   -3
  ----
13 | 2  2x-1
 -2
---
1 | 1    1x-1
1
--
0
Hence divisible by 11  Q =123

Eg: 1968 is divisible by 16

196| 8   Multiply by 1/6
 195| 18  8 is not divisible by 6 hence 1 borrowed
     3  
-----  
19| 2
18| 12
  2
---
1   6
1
--
0       Hence divisible .  Q = 6 12 18 divide by 6 Q=123      





TABLE  1  


Divisor
the last digit multiply with
11
-1   
12
- 1/2
13
- 1/3
14
- 1/4
16
- 1/6
17
- 1/7
18
- 1/8
19
- 1/9
21
-2   
22
-1   
23
- 2/3
24
- 2/4
26
- 2/6
27
- 2/7
28
- 2/8
29
- 2/9
31
-3    
32
--3/2
33
-1   
34
- 3/4
36
- 3/6
37
- 3/7
38
- 3/8
39
- 3/9
41
-4   
42
-2   
43
-4/3
44
-1   
46
- 4/6
47
- 4/7
48
- 4/8
49
- 4/9

Sum of the digit tricks:

Eg : 4567872  is divisible by 37

Sum of  3 digits from right ie 872+567+4 =1443 which is divisible by 37




Eg: 1222155 is divisible by 99

Sum of 2 digits 55+21+22+1 =99 which is divisible by 99

Divisor
3
9
11
27
33
37
99
101
Block to add
1
1
2
3
2
3
2
4


Sum of alternate digits :

Eg: 9012345597  is divisible by 73

Sum of alternate 4 digits ie 5597+90 =5687  next 1234 5687-1234=4453 which is divisible by 73

Eg: 12469056 is divisible by 101

Sum of alternate 2 digits = 56+46 =102 next 90+12 =102 102-102=0 hence divisible

Divisor
7
11
13
73
77
91
101
Block to add alternately
3
1
3
4
3
3
2
------------------------------------------------------------------------------------------------ 
                     Reference : 

1.      101 Shortcuts in maths an one can do by  Gordon Rockmaker-Feredrick Fell publishers,Newyork 1965
2.      Trenchenberg system of speed maths
3.      Secret of mental maths by Arthur Benjamin and Michael Shermer
4.      Vedic Maths
5.      Mental Maths Tricks by  Daryl Stephens
6.      Mental arithmetic tricks by Andreas Klein
7.      Simple Divisibility Rules by C.C.Briggs Penn State University
8.       Stupid Divisibility Test by Marc Renault
9.      Marvelous maths tricks


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Prepared by
O. MADHAVARAJ


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