Dedicated to my parents
Prepared by O. MADHAVARAJ
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
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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
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
| ||||
| ||||
6x3 =18
|
|
6x4 +5x3 =39+1 (reminder) =40
|
|
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
|
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 3 =1728
1728 1728x1/3 (1728x1/3)x1/3 [(1728x1/3)x1/3]x13 (a3 a2b ab2 b3 )
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)
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+32 =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
x2 +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 =342 =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
224 =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:
|
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
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
| ||||
| ||||
3x4 =12
| ||||
| ||||
3x3+2x4 = 17+1=18
| ||
|
3x2+2x3+1x4 = 16+1 =17
| ||||
| ||||
2x2+1x3 =7+1 =8
|
|
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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