Figuring Made Easy

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Number One Bestseller ller The Numbe

SHAKUNTALA DEVI THE MAT ATHEMAT ATICAL WIZARD

FiguringMADE EASY

DIS DISC O VER THE SEC RET O F  THIS MATHEMATIC AL SUP UPER ER STAR

Shakuntala Devi

Figuring Made Easy FOR TEENAGERS

w

Hind Pocket Books

FIGURING MADE EASY Copyright © Shakuntala DeVi, Hind Pocket Books

Published by HIND POCKET BOOKS (P) LTD. G. T. Road, Shahd ara, Delh i

110032

Pri nt ed in Indi a at

I. B. C.

PRESS

G . T . R o a d , S h a h d a r a , D el el h i 1 1 0 0 3 2

*'What's one and one and  one  one and  one  one and  one  one and  one  one and  one  one and  one  one and  one  one and  one?"  one?" "I don't know," said Alice, "I lost count." "She can't do addition," said the red  queen.  queen.

Lewis Carroll

CONTENTS

Introduction

9

1 T h e Family Fam ily of Numbe Num bers rs

11

2 Addition Addit ion

26

3 Subtrac Subt raction tion

35

4 Multiplic ation

39

5 Division

65

6 Some Special Spec ial Numb Nu mber erss

71

7 Games to Play with Nu mber mb er s

84

8 Puzzles to Puzzle You

89

 INTRODUCTION  INTRODUCTION

 Nu  N u mb e r s or orig igii nate na tedd hi t h e p r a c t i c a l need ne edss of h u m a n s . T h e  pr  p r i m i tiv ti v e h u m a n s b a r t e r e d a r tic ti c les le s w i t h o ne a n o t h e r o r divi di vidded their the ir prey into portions. And An d so as to maint ma int ain ai n a rough roug h  just  ju stic icee in t h e i r divi di visio sionn o r e xcha xc hang ngee , the th e y were we re fo forc rced ed t o deve de ve-lop some elementary notions of counting. Th e hu huma mann who fir first st realised th at t he numb nu mber er of fing finger erss on one hand ha nd or on two and an d the th e nu numb mber er of toes on the th e feet fee t were a convenient conveni ent tool for help he lp 'in counting coun ting must have ha ve been bee n considered a genius in his or her time. Since then th en there the re has been bee n an immense stride str ide in hu ma n though tho ughtt th at has carrie car riedd us, us, from fr om countin cou ntingg to the th e study of ' Nu mb er ' in itself itself as an object objec t of interest, intere st, and it daw ned on the human mind that numbers had an interest of their own. Th er e is is hardly har dly any activity in life which does not depend dep end on a cer tain tai n ability to count. coun t. We live in a world that th at is rul ed  by nu numb mber ers. s. Earl Ea rlyy in o u r chil ch ildh dhoo ood, d, t h e ' n u m b e r ' t h o u g h t  begi  be gins ns in us, when wh en we deci de cide de t h a t we woul wo uldd r a t h e r h a v e two tw o  piec  pi eces es of choc ch ocol olat atee than th an,, one o ne or we woul wo uldd r a t h e r h a v e m o r e than one toy. We see see numbers numbe rs and an d hear he ar the m every day at home. hom e. O u r houses hou ses have a num ber , we all have a birthd bir thday ay tha t ha t is is a num be  b e r , a n d those tho se of us who wh o h a v e tele te leph phon onee s kn know ow t h a t it has ha s also al so a number num ber—Da —Dadd has to be at work at nine o'clock, o'cloc k, mum mu m sets her he r oven tempe te mpe ratur ra turee at 320 320°F °F when she bakes a cake, big bi g sister sister has two pigtails, bab y br ot he r weighed nin e pound p oundss when he was bor born, n, my favour fav ourite ite T V pro progra gra mme is on channe cha nnell eight . . . Even family relationships involve a certain amount of counting . . . 3 sister sisters, s, 1 bro the r, 2 uncles uncles and so on.

10 Figuring Made Easy

Besi Beside dess the th e practical pract ical use nu numb mbers ers have, th ere er e is a range ran ge ol richness to nu numbe mbers; rs; they the y co come me alive, al ive, cease to be symbols written on a blackboard, and lead us into a world of arithmetical adventu adve nture, re, where calculating calculat ing numbers, and getting the th e right answers, can be exciting and thrilling. I fell in love with wit h number num berss at thre th ree. e. And I found it sheer ecstasy to do sums and get the th e right rig ht answers. Number Num berss were wer e my toys. toys. I played play ed with them. th em. My interes int erestt grew with wi th age and I took took deligh del ightt in working out huge problems probl ems mentally—sometimes even faster t han ha n electronic calculati calcu lati ng machines machines and computers. comput ers. And I feel, had ha d I been deprived depri ved of of my interest in numbers and the th e ability to calculate, calculate , the world would have  been  be en a du dull ller er plac pl acee for me. I see t h e worl wo rldd of nu numb mber erss as on onee of magic, full of fantasy and enchantment. In this thi s book I am taking taki ng you on an excursion into in to tiie tii e exciting excit ing world of numbers, number s, so we can share the th e adve ad vent ntur uree of attacking the problems in new ways, discovering different  possib  pos sibili ilitie ties, s, make ma ke u p new pr prob oble lems ms and an d expl ex plor oree t h e str st r ange an ge  prop  pr oper erti ties es of nu numb mber erss and an d the th e myste mys teri ries es of a r i t h m e t i c a l  phen  ph enom omen ena. a. First of all, let's meet meet the th e family of numbers. Th e family consists of ten te n number? num ber? and each one of them th em has a special character. We shall meet them one by one.

THE FAMILY OF NUMBERS

 NUMBER

1

 Nu  N u m b e r 1, or t h e u n i t y , is t h e basi ba siss of any an y system sys tem of n o t a tion. Upr igh t and unbend unbe ndin ingg nu numbe mbe r 1 has some some very special characteristics—the firs firstt and most most importa impo rtant nt being that th at whatwhat ever eve r figu figure re you multip mul tiply ly bv it, it , or divide divi de by it, it remain rem ainss un change cha nged. d. Th e firs firstt nine places in the th e 1-times table ta ble , the ref ore , have hav e no secret steps, or at least they the y a re t he same as th e normal ones, but after that, it becomes more interesting. 10 11 12 13 14 15 16

X X X X X X X

1 1 1 1 1 1 1

= = = = = = —

10 11 12 13 14 15 16

1 + 0 = 1 1 + 1=2 1+2 = 3 1 4- 3 = 4 1+ 4 = 5

1+5 = 6 1+6 = 7

an d so on. No ma tt e r how far fa r you go, you will find find th at th e Secret Secre t Steps always always give give you the th e digits fro from m 1 to 9 repe re peat atin ingg themselves in sequence, for example: Straight Steps

154 X 1 = 154 155 155 X 1 = 155

Secret Steps

1 + 5 + 4 = 10 1 + 5 + 5 = 11

1 + 0 = 1 1 + 1 = 2

12

Figuring Made Easy Secret Steps

Straight Steps

156 157 158 159 160 161

X 1 X 1 X 1 x' 1 X 1 X 1

= = = = = =

156 157 158 159 160 16 0 161

1 1 1 1 -1 1

+ + + + + +

5 + 5 + 5 + 5 + 6 + 6+

6 7 8 9 0

= 12 = 13 =14 = 15 =

1=

1 1 1 1 1

1+

+ + + + +

2 3 4 5 6

= = = = =

3 4 5 6 7

6 + 1 = 8

In this t his case, it is fai rly obvious why wh y the Secr et Steps wor k,  bu  b u t n o n e t h ele el e ss, ss , it is a curi cu rios osit ityy t h a t m a y no nott h a v e o c c u r r e d to you before. He re is an othe ot herr curiosity curiosity abo ut th e nu mb er 1 showing its tale nt for creat ing palind romes (num bers tha t read th e same  ba  b a c k w a r d s a n d f o r w a r d s ) 1 121

1 X 1

11 X 11 111 X  111 1111 X 1111 11111 X 11111 111111 X 111111 1111111 X 1111111 11111111 X 11111111 111111111

X

111111111

12321 1234321 123454321 12345654321 1234567654321 123456787654321 12345678987654321

At that point, it stops but the same thing works briefly with number 11: 11X11=

121

11 X 11 X  11 — 1331 11 X 11 X 11 X 11 = 14641

Th e fact th at 1 is, so to speak, im mu ne to mult ipli cati on means that whatever power you raise it to, it remains unchanged: 1564786 _ . J By the same sam e token ^ 1 = 1

The Family of Numbers

13

It is the only number all of whose powers are equal to itself. An d any nu mb er what wh atev ever er except zero raised to the th e zer o  po  p o w er is e q u a l t o 1. The powers exceeding 1, of all numbers greater than 1, are greater than the numbers so raised: 2s = 4 3* = 9 But the powers exceeding 1, of all quantities smaller than 1,  bu  b u t g r e a t e r t h a n 0 a r e s m a l l e r t h a n t h e q u a n t i t i e s : (1/2)2 = 1/4 (1/3)* = 1/9 Th e prod uct obta ined by multip lying to gether any two num bers, be rs, each gre ate r th an 1, gives gives a qu quan antit tit y lar ger th an either eit her of the th e multi pliers plier s and an d if the th e quanti qua ntitie tiess are less less th an 1,  bu  b u t g r e a t e r t h a n 0 , t h e i r p r o d u c t will wi ll b e a s m a l l e r q u a n t i t y than either of them. Th us , th e nu mber mb er 1 stands stan ds as a dividing dividin g point and an d qu quan an-tities greater than 1 present characteristics differing from those of quantities less in value than 1.

 NUMBER

2

 Nu  N u m b e r 2 has ha s o n e very ve ry ob obvi viou ouss c h a r a c t e r i s t i c — t o m u l t i p l y any nu numb mb er by 2 is the th e same as as add ing in g any numbe num berr to itself. itself. As for th e multip mul tiplica lica tion tio n an d its Secret Steps: 2x1 2X2 2X3 2X4 2 x 5 2X6 2X7 2X8

= 2 = 4 = 6 = 8 =1 0 =12 =14 =16

2

4 6 8

1 1 1 1

+ 0 +2 + 4 +6

= 1 = 3 = 5 = 7

14

Figuring Made Easy

2 x 9 2 2 2 2 2 2 2 2 2

=1 8

X 10 = X 11 = X 12 =

x 13 X 14 X 15 X 16 X 17 X 18

= = = = = =

1 +8 = 9 2 + 0 = 2

20 22 24 26 28 30 32 34 36

2 + 2 = 4 2 + 4 = 6 2 + 6 = 8

2 + 8 = 1 (0) 3 3 3 3

+ 0 = 3 + 2 = 5 + 4 = 7 + 6 = 9

and so on. The Secret Steps go on working  ad infinitum, always giving you th e same sequence of the th e fo four ur even digits followed followed by the th e five odd ones. He re is an amusing par ty trick th at can be played with number 2. The game is to express all ten digits, in each case using the nu mb er 2 five five times time s and no ot he r num ber s. You ar e allowed to use th e symbols symbols for ad di ti on , sub trac tion , multip licatio n and division and the conventional method of writing fractions. Here we are: 2 + 2 - 2 - 2 / 2 2 + 2 + 2 - 2 2 + 2 - 2 + 2/2 2/2 2 x 2 x 2 — 2 2 + 2 + 2-2/2 2 + 2 + 2 + 2 22-^-2 — 2 — 2

2 X 2 X 2 + 2 2 x 2 x 2 + 2/2 2 -

2/ 2 -

2/ 2

=1 - 2 = = - 2 = = - 2 =

2 3 4 5 6

=7 - 2 = 8 = 9 = 0

The Family of Numbers

15

Finally, here is another oddity associated with 2: 12345678 9 +123 45 67 89 + 9876543 2 1 + 987 65432 1

+2 22 22 22 22 2

 NUMBER

3

The most distinguished characteristic of this number is that it is the th e first first tri angle ang le nu mb er —t ha t is, is, a n um be r th e uni ts of which can be laid out to form a single triangle, like this 0%. Triangle numbers .have importance and peculiarities of their own which we shall encounter later on. Th re e is also also a pri me nu mb er. A prime nu mb er is one which whi ch is indivisible indiv isible exce ex cept pt by 1 and a nd by itself. 1, 2, 3, 5, 7, 11 11 an d 1133 are prime num ber s. Th e next pri me nu mb ers are 17, 19 and 23 and so on. Her e are some stra nge thing s abo ut th e firs firstt few pr im e numbers 1, 3, 5 : 153 = 1« + 5* + 3* An d 3 an d 5 can also be expressed as the th e diffe di ffe rence ren ce bet ween wee n two squares. 3 = 2* — I s 5 = 3* - 2*  Now  No w t h e Secr Se cret et S t e p s in t h e 3-ti 3- tim m es t a b l e : Straight Steps

3 x 1 = 3 3 X 2 = 6

Secret Steps

3 6

16

Figuring Made Easy Secret Steps 9 1+2 = 3 1+5 = 6 1 +8 = 9 2 + 1 =3 2 + 4 = 6 2 + 7 = 9 3 + 0 = 3 3 + 3 = 6 3 + 6 = 9

Straight Steps 3 X 3 = 9

3 3 3 3 3 3 3 3 3

X

X X

X X X

X X X

4 = 5 = 6 = 7 = 8 = 9 = 10 =• 11 = 12 =

12 15 18 21 24 27 30 33 36

Very Ve ry simple really. real ly. T h e pa tt er n of th e Secret St?ps recur s whatev wha tev er stage you car ry the th e tabl ta blee upto—why not check fo forr yourself 9

 NUMBER

4

Wi th n um be r 4, the Secret Steps in th e multiplic multi plication ation tables become a little more intricate: Straight Steps 4 X 1 = 4 4 X 2 = 8 4 X 3 = 12 4 X 4 = 16 4 X 5 = 20 4 X 6 = 24 24 4 X 7 = 28 4 X 8 = 32 4 X 9 = 36 4 X 10 = 40 4 X 11 11 = 4 4 4 X 12 = 48 4 X 13 = 52 4 X 14 = 56 4 x 15 = 60 4 X 16 = 64

Secret Steps

4 8

2 + 8 = 10

4 + 8 = 12 5 +

6=11

6 + 4 = 10

1 + 2 = 1 + 6 = 7 2 + 0 = 2 + 4 = 6 1 + 0 = 3 + 2 = 5 3 + 6 = 4 + 0 = 4 4 + 4 = 1+2 = 3 5 + 2 = 1 + 1=2 6 + 0 = 1 + 0 = 1

3 2 1 9 8 7 6

17

The Family of Numbers

At first, the sums of the digits look like a jumble of figu figure res, s, bu t choose choose at ra nd om any an y sequ ence of n umbe um bers rs an d mul tip ly them th em by 4 an d you will see that th at the pa tt er n emerges of two interli nked columns of digits in descendi desc ending ng ord er, for example: 2160 X 4 = 2161 X 4 = 2162 X 4 = 2163 X 4 = 2164 x 4 = 2165 X 4 = 2166 X 4 = 2167 x 4 =

8640 8640 8644 86 44 8648 86 48 8652 8656 8660 866 0 8664 866 4 8668 866 8

8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 +

6 + 4 + 6-1-4 + 6 + 4 + 6+ 5 + 6 + 5 + 6 + 6 + 6 + 6 + 6 + 6 +

and 2168 216 8 x 4 = 8672

0 = 4 = 8 = 2 = 6 = 0 = 4 = 8 =

18 22 26 21 25 20 24 28

2 + 8 = 10

8 + 6 + 7 + 2 = 23

 NUMBER

1 2 2 2 2 2 2

+8 = + 2 = + 6 = + 1 = + 5 = + 0 = + 4 =

9 4 8

3 7 2 6

1 + 0 = 1 2 + 3 =

5

5

Th e most impo im port rtan antt thin th in g abou t nu numbe mbe r 5 is th at it it is half of ten. This fact is a key to many shortcuts in calculation. Th e Secret Steps in the th e 5-times 5-time s table tab le are very similar to those thos e in the th e 4-times 4-time s tabl ta ble; e; the sequences sequence s simply go up upwar war ds instead of downwards: 5 5 5 5 5 5

=

X

1 2 3 4 5 6

5 X

7

=

5 X

8

=

X X X

X X

= = = =

=

5 X 9 = 5 X 10 = 5 X 11 = 5 X 12 = 5' X 13

5 10 15 20 25 30 35 40 45 50 55 60 65

5 1 1 2 2 3 3 4 4 5 5 6 6

+0 = + 5 = + 0 = + 5 = + 0,= + 5 = + 0 = + 5 = + 0 = + 5 = 10 + 0 = + 5=11

1 6 2

7

3 8 4 9 5 1+ 0 =

1 6

1+ 1 =

2

18

Figuring Made Easy

5 5 5 5 5 5

X X X X X X

14 15  \6 17 18 19

=

= = =

= =

7 7 8 8 9 9

70 75 80 85 90 95

+ + + + + +

0 5 0 5 0 5

= = 12 = = 13 = = 14

7 3

1 + 2 = 8

4

1 + 3 = 9 1 + 4 =

5

and so on.

 NUMBER

6

This Th is is th e second triang tri ang le nu mb er ; an d the th e firs firstt per fect fec t numb nu mber er—a —a perf p erfect ect nu mber mb er is is one which is equal to the sum of all its diviso divisors. rs. Th us 1 + 2 + 3 = 6. T h e Secret Steps in the 6-times 6-ti mes table tab le are very similar to those in the 3-times table, only the order is slightly different. Straight Steps 6 6 X 1 = 6 x 2 = 12 6 X 3 = 18 6 X 4 = 24 6 X 5 = 30 6 X 6 = 36 6 X 7 = 42 6 X 8 = 48 6 X 9 = 54 6 X 10 = 60 6 X 11 = 66 6 x 12 = 72

Secret Steps

4 + 8 = 12

6 + 6 = 12

1 1 2 3 3 4 1 5 6 1 7

+ 2 +8 + 4 + 0 + 6 + 2 +2 + 4 + 0 +2 + 2

=6 = 3 = 9 = 6 = 3 = 9 = 6 = 3 = 9 = 6 = 3 = 9

and so on.

 DUMBER

7

This is the next prime number after 5. The Secret Steps in the 7-times table almost duplicate those in th e 2-times, except excep t th at they t hey go up instead of of down at each step:

19

The Family of Numbers Straight Steps

Secret Steps

7 X

1

=

7 X 7 X

2

=

7 14

3

=

21

7 X

4

=

28

7 X

5

=

35

3 +

5 =

8

7 X

6

=

42

4 + 2 =

6

7 X

7

=

49

4 + 9

7 X

8

=

56

5 +

9 = 7 X 10 = 7 X 11 =

63

7 X

7 1 * 4 - 5 2 + 1 = 3 2 + 8

1 + 0 = 1

1 + 3 = 4 1 + 1 = 2

= 13

6 = 11

6 + 3 = 7 + 0 =

70 7 + 7 = 14 8 + 4 = 12

77

7 X 12 = 8 4 7 X 13 = 91 7 X 14 = 98 7 X 15 = 105 7 X 16 = 112 1 +

1 + 4

= 5

1 + 2

= 3

1 + 7 = 8 1 + 0 + 5 = 6 1 + 1 + 2 = 4 1 + 1 = 2

1 + 9 2 = 11

1 + 2 + 6 = 9 1 + 3 + 3 = 7 1 + 4 + 0 = 5

= 140

There is a curious relations hip between between 7 and the 142857.

Watch

7 x 2 7 X 2

=

2

7x 2*

=

7 X 2

4

7 X 2

s

7 X 2« 7X2' 7 X 2

=

8

7 X 2»

=

= = = = =

7 X 7 X 7 X 7 X 7 X

9 7

1 + 0 = 1

9 + 1 - 10 9 + 8 = 17

7 X 17 = 119 7 X 18 = 126 7 X 19 = 133 7 X 20

= 10

2 4

= 14 28 =

8 = 16 = 32

=

7 X 64 = 7 X 128 = 7 X 256 25 6 = 7 X 512 =

56 112 224 22 4 448 896 1792 3584 142857142857142(784)

nu mb er

20

Figuring Made Easy

 No  N o m a t t e r ho how w f a r yo youu t a k e t h e c a l c u l a t i o n , t h e sequ se quen ence ce 1428 14 2857 57 will repe re peat at itself thou th ough gh th e final fina l digits dig its on the th e righ rightthand ha nd side whic h I have bra cke ted will will b e 'wron 'w ron g' becaus e they would be affected by the next stage in the addition if you took the calculation further on. Th is n um be r 14 1428 2857 57 has ha s itself some strange stra nge pro proper perties ties;; mul tip ly it by any nu numb mber er betwe en 1 and 6 a nd see see what wh at happens: 142857 142857 142857 142857 142857 142857

x 1 x 2 X 3 X  4 x 5 x 6

= = = = = =

142857 285714 285 714 428571 571428 5714 28 714285 7142 85 85714 85 7142 2

T h e same digits recur recu r in each eac h answer, and if th e prod ucts ar e writ wr it ten te n each ea ch in the th e fo form rm of a circle, circ le, you will will see see th at th e ord er of th e digits remains rema ins the same. If you y ou th en go on to mult mu ltip iply ly t h e same nu mb er by 7, th e answer is 999 999999 999.. We will come back to some fu rt he r char acte rist ics of this nu numb mb er in a later chapter.

 NUMBER

8

Thi s tim e th e Secret Steps in th e multip licatio n tabl e are the reverse of those in the 1-times table: Straight Steps

8 X 1 = 8 8 8 8 8 8 8 8

X X X X X X X X

2 3 4 5 6 7 8 9

=» = = = = = = =

Secret Steps 8

8 16 24 32 40 48 56 64 72

1+6 2 + 4 3 + 2 4 + 0 4 + 8 = 12 5 + 6=11 6 + 4 = 10

= = = =

7 6 5 4

1+2 = 3

1 + 1=2 1 + 0 =1 7 + 2 = 9

The Family of Numbers

8 8 8 8 8 8 8 8

X 10 = X 11 = X 12 = X 13 = X 14 =

X 15 = X 16 = X 17 =

80 88 96 104 112 120 128 136

8 + 8 = 16 9 + 6- = 15

1 + 2 + 8 = 11 1 + 3 + 6 = 10

21

8 + 0 = 8 1 + 6 = 7 1 +5 = 6 1 + 0 + 4 = 5 1 + 1 + 2 = 4 1 + 2 + 0 = 3 1 + 1 =2 1 + 0 = 1

So it continues. If this is une xpe cte d, then the n look look at some othe r peculiarities of the number 8: 888 88 8 8 8

1000

and 88 888 8888 88888 888888 8888888 88888888

= = = = = = =

9 98 987 9876 98765 987654 9876543 98765 43

X 9 + 7 X 9 + 6 X 9 + 5

X x X x

and lastly 123456789 X 8 =-987654312

9 + 4 9 + 3 9 + 2 9 1

22

Figuring Made Easy  NUMBER

9

With 9 we come to the most intriguing of the digits, indeed of all num ber s. T h e num n um be r 9 exceeds exceeds in intere sting and an d  pr  p r a c t i c a l p r o p e r t i e s of all al l o t h e r dig di g its: it s: Straight Steps 9 x 1 = 9 9 X 2 = 18 9 X 3 = 27 9 x 4 = 36 9 X 5 = 45 9 x 6 = 54 9 X 7 = 63 9 x 8 = 72 9 x 9 = 81 9 x 10 = 90 9 x 1 1 = 99 9 X 12 12 = 108

1 + 8 = 9 2 + 7 = 9 3 + 6 = 9 4 + 5 = 9 5 + 4 = 9 6 + 3=9 7 + 2 = 9 8 + 1=9 9 + 0 = 9 1+ 8 = 9 1 + 0 + 8 = 9

9 + 9 = 18

It is is an absol ute rule th at what wh at ever ev er you multip mul tiply ly by 9, t h e sum of t he digits in th e pr prod oduc uctt will always be 9 pro provid vid ed this thi s sum consists of one digit only. only . If th e sum of t he digits digi ts consists consists of mo re than th an one dig it, it , go on ad di ng the digits of th e sum til l you get a one-digit one-di git nu mb er , as shown in the Secret Secr et Steps Step s above. above . You will find find t h a t this one-digit one-di git nu numb mber er will always be 9. More Mo reov over er,, not only ar e th er e no steps in th e hi dd en pa rt of the 9-times tab le bu but, t, for its firs firstt ten places, it has another feature: 1 X 9 2 X 9 3 x 9 4 x 9 5 x 9

= = = = =

09 18 27 36 45

90 = 9 81 = 9 72 = 9 63 = 9 54 = 9

X X X X X

10 9 8 7 6

T h e pro duc t in the second half of the th e table tab le is the th e reverse revers e of t h a t in t h e firs first. t.

The Family of Numbers

23

 Now,  Now , t ake ak e a ny n u m b e r , say, sa y, 87 8759 594. 4. R e v e r s e t h e o r d e r of the th e digits, wh ich gives you 49578. 49578. Su bt ra ct th e lesser fr om the greater 87594 - 49578 38016  Now  No w a d d u p t h e s u m of d igi ig i ts in t h e r e m a i n d e r : 3 + 8 + 0 + 1 + 6 = 18

1+8 = 9

The answer will always be 9. Again, take any nu mb er , say say 64 6478 783. 3. Cal cul ate the sum of its digits: digit s: 6 + 4 + 7 + 8 + 3 = 28. (You can stop th er e or go on as usual to calc ca lc ul ate at e 2 + 8 = 10 10,, an d ag ain ai n only if you w i sh , 1 + 0 = 1 . )  Now  No w t a k e t h e s u m of t h e d i g its it s away aw ay f r o m t h e o r i g i n al num ber , and add up the digits of the rem ain der . Wher ever you choose choose to stop, an d wha tev er the nu mb er you originally select, the th e answer will bp 9 if th e sum of th e digits of th e re main ma in der de r is a one-digit nu mb er , otherwi se this thi s sum will will be a multiple of 9. 64783 -

28

64755 647 55

6 + 4 + 7 + 5 + 5 = 27

2 + 7 = 9

6 + 4 + 7 + 7 + 3 = 27

2 + 7 = 9

6 + 4 + 7 + 8 + 2 = 27

2 + 7 = 9

64783 -

10

64773 64783 - 1 64782 647 82

24

Figuring Made Easy

Take the nine digits in order and remove the 8: 12345679.

Then multiply by 9: 12345 123 45679 679 x 9 = 111

111 111

111

 Now  No w t r y m u l t i p l y i n g by t h e m u l t i p l e s of 9: 12345679 12345679 12345679 12345679 12345679 12345679 12345679 12345679

X X X x X X x x

18 27 36 45 54 63 72 81

= = = =

= = = ==

222 333 444 555 666 777 888 999

222 333 444 555 666 777 888 999

222 333 444 555 666 777 8§8 999

Th er e is a little mat hem ati cal rid dle th at can be played using 9. What is the largest number that can be written using three digits? 99

T h e answer answe r is 9 or 9 raised to th e ni nt h power ol 9. T h e ni nt h power pow er of nine ni ne is 38742048 387420489. 9. No one knows the th e precise n u m b e r th a t is r ep r es en t ed by b y 9»8742«48» j b u t it begins 42812477 3...and ends 89. Th e compl ete num ber will will contain 369 milli mi lli on dig its, its , would wou ld occupy occu py over ove r five five hu nd re d miles of  pa  p a p e r a n d t a k e year ye arss to r e a d .

 ZERO

I have hav e a pa rt icul ic ul ar affection for zero because it was some of my coun co un trym tr ymen en who first first gave it th e status sta tus of a nu numb mb er. er . Th ou gh the symbol for a void or nothingne ss is ^h^ugh ^h^ ugh t to have hav e bee n invente inv ente d by the Babyl onians,^l onia ns,^l t was* tEe Hi nd u ma th em atic at ic ia ns who' who' firs firstt conceived of 0 as a numbe num be r, the next in the progression 4-3-2-I.TW l Mftycv Mftycvr^ r^ a l s o mv«*vV 126 23-> 252 ll->- 504 5 - * 1008 1008 2->-2016 l->4032

b 92 d a mm i be r by 5, yo youu t ake ak e ad v a n t ag e of th e fa ct that. is HaH HaH'of 'of 10, mult mu ltip iply ly the divi di viden den d (the (t he nu mber mb er being bein g di vidt vi dt c) ty t y 2 and an d divide divid e by 10 by moving mov ing the th e decimal point one place pla ce to the th e lef t. For example exa mple:: 1655 divi 16 di vide dedd b,b,- 5 =

330

_ 33

i o "-•'.Jc "-•'.Jc by 15 15.. multip mul tiply ly the th e d.vi d. vide dend nd by 2 and divide by 'V;.  l - ' . ; r   {xaiv.ple:

66

Figuring Mad* Easy

divid di vid end en d — an d ar e left lef t at the end of th e calculat calc ulation ion with w ith a remainder, you must remember that this, too, will be doubled. Some divisions can be simplified by hal vin g bo th divisor divisor and divi di vide dend nd — divisions divis ions by 14 14,, 16 16,, 18 18,, 20, 20, 22 a n d 24 becom be comee divisions by 7, 8, 9, 10, II and 12, respectively — but this time you must re mem be r to doub le the r emai nder . He re are some examples: 196 392 divi di vide dedd by 14 =

= 28 232

464 divided by 16 =  ~

= 29

882 divi di vide dedd by 18 = 441 4960 divi di vide dedd by 20 = ~

49 = 248

473 946 divi di vide dedd by 22 = -  j j -

43

11766 divi 117 di vi de d by 24 = 588 = 49  Dividing by Factors

If a nu numbe mbe r can be brok en down into fac tor s, it ma y b e simple sim plerr to divi di vide de by these, the se, successively, successively, t h a n to do a single calculat calc ulation. ion. A me nt al division division by 8, an d th en by 4, is simpler simp ler th an a division by 32. For exam e xam ple, ple , to div ide 108 10888 by 32, fir first st divide by 8: ^ = 1 3 6 and then divide the answer, 136, by 4 to get the final result—34.  Nu  N u m b e r s easy eas y to h a n d l e in this th is way wa y a r e t h e p r o d u c t s in t h e  basi  ba sicc m u l t i p l i c a t i o n t a b l e s — m u l t i p l e s of 11, f o r i n s t a n c e , go  pa  p a r t i c u l a r l y s m o o t h l y : To divide 2695 by 55, first divide by 5: _ 539

 Division

67

and then by 11: 535 11

49

to get the answer. T o divide div ide by nu numbe mbe rs that th at are ar e powers of 2 (4, 8, 16, 16, and an d so on), you merely mer ely have to go on hal vin g the dividend. divide nd. 16 16,, for instan ins tance, ce, is 2*, so halvi ha lving ng the th e divid di viden endd fo four ur times is the th e same t hi n g as divi di vidi ding ng by 16. 16. To divide 8192 by 16 halv ha lvee once onc e to get and an d again aga in to get and an d a th ir d time tim e to get and, an d, final finally ly,, a fou fourth rth time to get ge t the th e answer

4096 4096 2048 1024 10 24 5122 51

Th is techn tec hniq ique ue makes dividing divi ding by high powers powers of 2 easy, for instance, to divide   32768   by 128 — which is 2 7:

an d

32768 halv ha lved ed 16384 16384 halv ha lved ed 8192 halv ha lved ed 4096 halved hal ved 20488 halv 204 ha lved ed 1024 h a l v e d 512 halve hal vedd

« = =* =* — = = =

163844 1638 81922 819 4096 2048 1024 10 24 5 1 2 256, which whi ch is th e requi re quired red answer.

Th e meth me thod odss I have described descri bed so f a r work only with some numbers numbe rs — thinking thin king about what sort of numbers numb ers ar e involved in a calcu ca lculat lation ion i$ of te n th e firs firstt step ste p to find findin ingg a quick way to do it. T h e more mo re skilled you get at ment me ntal al multiplic multi plication ation and divisi division, on, the mor e steps in a nor normal mal division sum you will be able to do ill your head. For instance, in the sum set out below, only the remaind rema inders ers are ar e noted down  —pp a r t i a l p r od  — oduu c ts a r e arri ar rive vedd a t , a n d t he subt su btra ract ctio ions ns d o n e mentally:

68

Figuring Made Easy l

31 ) 13113 ( 423 71 93 00  Dividing By Fractions Mentally

I t is easier easier to divide divi de by whole num ber s t h a n by fra ctio ns  — if b o t h divi di viso sorr a n d "d "div ivid iden endd a r e m u l t i p l i e d b y t h e same sa me factor, fac tor, th e answer to the t he pro ble m will will be the th e same. (Any rem ai nd er , however, will will b e a mult iple or f rac tio n of th e correc cor rectt value.) value .) If , for instan in stan ce, you are div idi ng by 7$, it is simple r to multip mul tiply ly bot h nu mber mb erss in the calcula tion tio n by 4 — dividing four times your original dividend by 30. For example, to divide 360 by 360 X 4 1\ X 1440 30

= 4 =

1440 30 48

By the th e same pri nci ple when dividing dividin g by 12J, multip mul tiply ly th e div idend ide nd by 8 and an d divide div ide by 10 100, 0, when div d ividi idi ng by 37$, mul tip ly by 8 and an d divide by 300 300,, and when div iding id ing by 62$, multiply by 8 and divide by  500. To divide by I J, doub le th e divi dend and th e divisor divisor and divide by 3 (but remember to halve any remainder). T o div ide a n u m b e r by 2J, do ub le th e di vi de nd an d th e div isor an d di vi de by 5 — a n d go ab ou t di v id in g b y 3$ in t h e same Way.

Chicking Whether a Number is Exactly Divisible

Th er e are tests th at can be ma de to show show whet wh ethe herr a num be  b e r is e xact xa ctly ly divi di visi sibl blee by a n o t h e r n u m b e r (or (o r a m u l t i p l e of it). Here are some of them: If a number  is  divisible divi sible by two, it will end in an even number or a zero.

 Division

69

If a n u m b e r is divisible by th re e, th e sum of its digits will  be  b e d ivis iv isii ble bl e b y 3 (for (f or e x a m p l e , 372 = 3 + 7 + 2 = 12). A corolla ry of this is is th at any nu mb er ma de by rear rang ing t h e digits dig its of a n u m b e r divisible divisi ble by 3 will also be divisible divisib le  by 3. If a nu mb er is divisible by four four,, th e last two digits a r e divisible divisi ble by 4 (or ar e zeros). zeros). If a number is divisible by five, the last digit will be 5 or 0. I f a n um be r is divisibl divi siblee by six, th e last digit digi t will be even eve n and the sum of the digits divisible by 3. T h e r e is no qu quic ickk test fo forr find findin ingg out if a n um be r is divisible by seven. If a n um be r is divisible by eight, th e last thre e digits are divisible by 8 or are zeros. If a n u m b er is divisible by nine, nin e, th e sum of its digits is divisible by 9. If a number is divisible by ten, it ends with a zero. If a nu mb er is divisible by eleven, the th e differen ce between t h e sum of th e digits in th e even places pla ces an d the th e sum of th e digits dig its in th e od oddd places plac es is a mu lt ip le of 11 or it is 0. For Fo r example, 58432 is shown to be divisible by 11 because: 5 + 4 + 2 = 11 11 8 + 3 = 11 I I — 11 = 0 and 25806 is shown to be divisible by 11 because: 2 +

8 + 6 = 16 5 + 0 = 5 16 — 5 = 11

Checking By Casting out the Nines

Divisions, like additions and multiplications, can be checked by casti ng ou outt the th e nines — alt hou gh, again, agai n, the met ho hodd is not infallible. You You obt ain check numb ers as befo re, by addi ng up th e digits digi ts in the th e n umbe um bers rs in th e calcu lati on—in on— in this case th e divisor, div ide nd and an d quot ient — an d dividing divi ding them th em by 9.

70

Figuring Made Easy l

The check number is the remainder left after each division. Mu ltip lt iply ly the th e check n u mb er of the divisor divisor by the th e check num ber be r of the th e quo tient. tie nt. If the check nu numb mb er of this pr od uct uc t is th e same as th e check nu mb er of the dividend divi dend , the th e division division ma y be assumed to be correc cor rec t. For example, examp le, to check: 24263762426376... - 5 3 2 F =  4 5 6 2 + 4 + 2 + 6 + 3 + 7 + 6== 6= = 3ff 3ff,, castin cas tingg ou outt th e nine ni ne s leaves 3. 5 + 3 + 2 + 1 = 11 11,, castin cas tingg ou outt th e nine? nin e? leaves 2. 4 + 5 + 6 = 1 5 , casting casting out the nines nines leaves 6. number divisor  X check check nu mb er quotient). Cast Ca stin ingg out th e nines ni nes leaves 3 — which wh ich is th e same sam e as th e check number of the dividend. W he n ther th eree is a re ma in de r, the sum of th e digits in the re mai nd er is is add ed t o the pr odu ct of of the check check nu mb er of the quo tie nt and th e divisor, and an d the th e operation oper ation is comp leted lete d in the usual usua l way. For examp exa mple, le, to check:

6 x 2 =

1 2  (check

4281

= 34 3466 re main ma in der de r 39.

1 + 4 + 8 + 1 + 2 + 6 + 5 = 2 7 c a s t i n g o ut ut leaves 0 4 + 2 + 8 + 1 = 15, cast castin ingg out leaves 6 3 + 4 + 6 — 1 3 , casting casting ou t leaves 4 3 + 9 = 12 ,  casting out leaves 3

th th e nines the

nine niness

th e

nines nines

the

nines

6 X 4 = 24, 24 , 24 + 3 = 27, ad ding di ng these digits an d cast ing

ou t the th e nnines ines leaves 0, whi ch is th e check nu mb er of th e dividend.

I

\

6 SOME SPECIAL NUMBERS

The Resolving Number — 142857

While discussing nu numbe mbe r 7 in Ch Chap ap ter te r 1, I described some some oddities odditie s of the nu mber mb er 14285 142857. 7. For instanc inst ance, e, its habit hab it of re peat  pe atin ingg itse it self lf in t h e t o ta l when wh en t h e po powe wers rs of 2 a r e mult mu ltip ipli lied ed  by 7 a nd a d d e d a f t e r b e ing in g set se t o u t in a sta st a gg ggee red re d fo form rmat atio ion, n, for instance. He re are ar e some some more. First Firs t set set out the dieits in a circle:

7

4

5

2

 Now mult mu ltip iply ly 142857 by n umb um b e r s f r o m 1 to t o 6:

72

Figuring Made Easy

l

142857 X 1

142857 X 2

142857

142857

285714

428571

X 3

142857 X4

142857 X5

571428

714285

142857 X 6 857142 You will see see th at the t he nu mber mb erss start sta rt revolving revol ving — th e same digits in different comb inat ions arrived at by star ting fro m a different point on the circle. Multiply 142857 by 7 and things suddenly change 142857 X 7 999999 But ther th eree are still odd odditie itiess in store, for instance inst ance,, if you multiply 142857 by a really big number, see what happens: 142857 X  32284662474 4612090027048218  No  N o r e s e m b l a n c e t o 142857 14285 7 a t first sigh si ghtt p e r h a p s , b u t div di v ide id e th e p ro rodu duct ct up in to groups of of 6, 6 and an d 4 an d see w hat happens: 048218 090027 4612 142857

Some Special Numbers

73

Th e revol ving nu mb er appears again: Sometimes 14 1428 2857 57 liides d ee per th an th at . For instance, in this multip licatio n, t h e pr od uct seems seems immu ne fro m 14 1428 2857 57 extr actio n: 142857 X 45013648 6430514712336 But divi di vide de it in to 6, 6 an d 1 in th e same sa me way wa y as you did last time and you arrive at the following sum: 712336 430514 6

1142856 Treat this total in the same way: 142856 1 142857 -And you winkle 142857 out. You ca n find find ou outt ooddit ddities ies ju st by lookin l ookingg at the t he digits digi ts themselves. For Fo r instance inst ance,, if yo youu divide divi de th em int o two groups, groups , 1422 an d 857, th e second 14 seco nd figure figure of th e firs firstt gr grou oup, p, mult mu ltip ipli lied ed  by t h e t h i r d figure of t h e first g r o u p give gi vess t h e first figure of t h e second group: 4 x 2 = 8

T h e sum su m of the th e first first two figu figure ress of t h e firs firstt gr ou p gives the th e second secon d figu figure re of t h e second seco nd gr grou oup: p: 1+4 = 5

74

Figuring Made Easy l

And An d the t he sum of all the th re e figures of the th e firs firstt gr group oup gives the third figure of the second group: 1 + 4 + 2 = 7 T o sho w, the »e xt pr op erty er ty of 14 1428 2857 57 you hav e to dr aw xip xip a tab le of the pro produc duc ts of the th e nu mb er when multip lied by 1, 2, 3, 4, 5, and an d 6. Hori Ho rizo zont ntal ally ly and an d vertical vert ically, ly, th e digits digit s all add up to 27: 1 2 4 5 7 8

4 8 2 7 1 5

2 5 8 1 4 7

8 7 5 4 2 1

5 1 7 2 8 4

7 4 1 8 5 2

27

27

27

27

27

27

=

= =

= =

=

27 27 27 27 27 27

And if you can' ca n' t re memb me mb er th e nu mb er , ther th eree is always an easy way to find it; for  1 /7, expressed as a decimal is: 142857 142857 142857 142857 142857 an d so on to infi in fini nity ty.. The Number 1089

some pecu liar tra its . Fo r This, as I mentioned earlier,, has some example, look look at th e pa tt er n t hat is formed when it is multi pli  p liee d b y t h e n u m b e r s 1 to t o 9: 10 89

98 01 =

1089 X 9

2178

87 12 =

1089 X 8

108 9 X 3 =

32 67

762 3 =

108 9 X 7

108 9 X 4 =

4356

65 34 =

1089 X 6

10 89 X 1 = 1089 X

2 = 

1089

x

5 =

5445

Or try this trick; write   1089 on a piece of pa pe r an d put pu t it in your pocket. Now ask someone to think of a three-digit num be  b e r , t h e first a n d last la st digi di gits ts of w h ich ic h diff di ffer er b y at leas le astt 2. L e t us suppose he chose 517. Now ask him to reverse the digits and 198) subtract the higher number from the lesser   (715 — 517 = 198)

Some Special Numbers

75

Finall y, ask hi m to add this nu mb er to itself itself reversed (1 98 + 891 = 10 1089 89.) .) No ma tt er what wha t nu mb er he starts star ts with, you will always have the final answer — 1089 —   in your pocket! Strange Addition

1 2

6

3 4 5 6 7 8 9

9 12 15 18 21 24 27

15 20 25 30 35 40 45

28 35 42 49 56 63

45 54 63 72 81

66 77 91 88 104 120 99 117 135 153

In the t able above, the horizontal lines are arit hmet ical  prog  pr ogre ress ssio ions ns.. T h e d i f f e r e n c e b e t w e e n e a c h n u m b e r a n d t h e one to its righ ri ghtt is twice twic e the th e fig figur uree th at stands sta nds at the th e be gi nn in g of the th e row. (Row 7, fo forr exampl e, can be worked work ed out by adding 14, first to 7 and then to each successive total.) But how would wou ld yo youu fin findd th thee sum of all th e nu mber mb er s in any an y row? Th er e is no need ne ed to add th em — it is the same as th e cube of the nu mb er which whi ch stands at th e beginn beg inn ing of the th e row. For instan ins tance, ce, th e tot al of th e nu mb er s in row 6 is 6 X 6 X 6 = 216. 216.

 Number 37

Watch the pattern this number forms when multiplied by 3 and its multiples: 37 37 37 37 37 37 37 37 37

X 

3=111 X 6 = 222 X 9 = 333 4444 X  12 = 44 X 15 = 55 5555 X 18 = 66 6666 x 21 = 777 X  24 2 4 = 888 X  27 = 999

76

Figuring Made Easy l

 Number

65359477124183

Wat ch the pa tt er n for med when you multip ly this num ber  by  b y 17 a n d m u l t i p l e s of 17: 65359477124183 65359477124183 65359477124 65359477124183 183 65359477124183 65359477124183 65359477124183 65359477124183 65359477124183 65359477124183

x x

x X X x x  x x

17 = 34 = 51= 68 = 85 = 1022 = 10 119 = 1366 = 13 1533 = 15

1111111111111 11111 11111111111 111 2222222222222222 22222222 22222222 33333333333 3333333333333333 33333 4444444444444 4444444 444444444 444 5555555555555555 555555 5555555555 666666666666666 66666666 666666666 77777777777 777777 7777777777 77777 8888888888 88888 8888888888 8888888 99999999999 999999 9999999999 99999

 Reversals

9 24 47 497 The Number

+ + + +

9 3 2 2

= 18 = 27 = 49 = 499

81 72 94 994

= 9 x = 24 X = 47 X = 497 x

9 3 2 2

526315789473684210

Thi s deceptively strai ghtf orwa rd nu mb er is also also circular, in the th e sense sense th at when whe n it is multipli mult iplied ed by any nu mb er fr om 2 to 200 200,, the prod p rod uct can always be read rea d off off fr om a circle ma de u p of the th e figures figures of th e mul tip lic and . If the th e multip mul tiplie lierr is 18 or less, less, the br break eak comes after af ter the th e di git gi t which is th e same as the th e multip mul tip lie r or or,, where whe re the multimu lti pl  p l i e r h a s two tw o figures, t h e s a m e as t h e last la st figure of t h e m u l t i  pl  p l i e r . W h e n t h e c u t h a s b e e n m a d e a n d t h e t w o p a r t s j o i n e d , add a zero to get the final answer. As all th e digits digit s in t he nu mb er , except 9 an d 0, occur occu r twic e, you must know which of the th e pair pai r to mak e the bre ak af te r. T h e ru rule le is this: th is: look at the th e fig figur ures es,, following follow ing th e two you are ar e choosing choosing between bet ween . If the th e nu mb er you are mul tiplying tipl ying  by  b y is bet be t wee we e n 2 a n d 9, m a k e t h e b r e a k b efo ef o r e t h e lowe lo werr figure, if it lies between betwe en 11 11 an d 18 make ma ke the th e br break eak bef or oree t he high hi gher er one. For example:

Some Special Numbers

77

To multiply 526315789473684210 by 6. T h e br break eak mus. mus.tt come betwee bet weenn 6 an d 3 or 6 an d 8. T h e multiplier lies between 2 and 9, so we make the break between 6 and 3. We write wr ite out th e answer by tak ing the figures th at follow 6 : 315 315789 789473 473684 684210 210,, carryi car rying ng on fr om th e beginning beginn ing of the th e number and joining on 526, and adding 0 to get the answer: 3157894736842105260 Or to multiply 526315789473684210 by 14: The break can be made between 4 and 7 or 4 and 2. 14 lies  betw  be twee eenn 11 a n d 18, so we choose cho ose t h e h i g h e r n u m b e r , a n d t h e answer can be read straight off: 7368421052631578940 To multi mu lti ply by numb ers between bet ween 19 a nd 200 is a litt le more comp licat ed. If they are th e num ber s we have already dealt with multiplied by 10—that is 20, 30, 40 and so on upto 180—there 180—there is no pro ble m, for we can carr y out the pro cedure ced ure set set out above an d add 0 to the produ ct. Simil arly, to mult iply  by 200 200,, c a r r y o n t h e o p e r a t i b n as if m u l t i p l y i n g b y 2 a n d a d d two zeros to the th e answer ans wer.. 19,, 38. 57, 76 and 19 an d 95 give pr od oduc ucts ts in whic wh ichh all the th e digits dig its are ar e nines nine s exce ex cept pt th e last two and an d (in all cases except exc ept 19) 19) t h e first:

78

Figuring Made Easy l

526315789473684210 X 19 9999999999999999990 526315789473684210 X 38 19999999999999999980 526315789473684210 X 57 299999999999999999970 526315789473684210 X 76 39999999999999999960 526315789473684210 X 95 49999999999999999950 These five numbers are special cases. The rule for numbers  be  b e t w e e n 21 a n d 29 is thi th i s: a d d 1 to t h e seco se cond nd d i g i t of t h e mult ipli er. Multi ply the special num ber by maki ng the break break  be  b e f o r e t h e lowe lo werr of t h e two tw o po possi ssibl blee figures. W h e n y o u r e a c h th e last digit , red uce uc e it by 1. 1. Ins ert 1 at th e begin be gin ning ni ng and 0 at th e end of the th e nu mb er to arrive arr ive at the th e fina finall pr prod od uct. uc t. For example: 526315789473684210 X 27 Incr ease the second second digit of the mul tip lier by by. 1 : 7 + 1 = 8 .  No  N o w m a k e a c u t b e t w e e n 8 a n d 4 in o u r s p ecia ec iall n u m b e r a n d Write do down wn the th e figure figuress 421052631578947367

Some Special Numbers

79

.Annex a zero to this nu mb er and an d at ta ch the 1 at the beginning to get the product 14210526315789473670 To mult iply the special nu mb er by num ber s between 31 and an d 37, use th e same method met hod as th at set out above, bu butt make ma ke t h e bre ak before bef ore th e highe hi ghe r of the th e two possible figu figure res. s. For Fo r example: 526315789473684210 X 34 In cre ase th e second Incre second digit of of th e mu ltip lt ip li er by 1 to m ak e 5. Ma k e t he bre ak betwee be twee n 5 and an d 7 to get the nu mb er 78947368421 78947368421052631 0526315. 5. Redu Re du ce th e last digit b y 1, pu t a 1 at th e  be  b e g i n n i n g a n d a 0 a t t h e e n d to a r r i v e a t t h e final p r o d u c t : 17894736842105263140

For multipliers between 39 and 48, the method is the same as for nu mb er s between bet ween 21 and an d 29, b ut instea d of  1  at the  be  b e g i n n i n g of t h e p r o d u c t , wri wr i te 2, i n s t ead ea d of d e d u c t i n g 1 fr from the th e last figu figure re of the nu mb er, er , de du ct 2, an d instead instea d of ad di n g 1 to the second digit of the multiplier, add 2. For example: 526315789473684210 X 46 Add 2 to th e second figu figure re of th e mult mu ltip ipli lier er:: 6 + 2 = 8. Make Ma ke a cut cu t at the th e lower succeedin succe edingg figu figure re.. You get: 421052631578947368 Reduce the last figure by two units. You get: 421052631578947366 An ne x a zero to this nu mb er and at ta ch the nu mber mb er 2 at the  be  b e g i n n i n g of t h e n u m b e r t o o b t a i n t h e p r o d u c t : 24210526315789473660

80

Figuring Made Easy l

Wh en th e pr prod od uct uc t lies between bet ween 49 and an d 56, th e metho me tho d is the same as th at for multiplie rs between 39 and 48 48,, bu t th e cut cu t is is ma de befor be foree th e hig her he r of the th e two possible possible figu figure res. s. For multip liers between betw een 68 and 75, 75, the me th od is as tha t for 58 to 67, 67, bu t th e brea br ea k is ma de at a t th e high hi gher er figu figure re.. For multipliers between 77 and 85, insert 4 and not 3, and cut at the lower figure and for multipliers between 86 and 94„ do the same but cut at the higher one. For multipliers from 96 to 104, insert ins ert 5 no t 4, an d cut at th e lower figu figure re.. For ot he r numbers, similar methods apply — you can find them by trial and an d error err or,, right rig ht upt o 200. You will find find thou th ough gh th a t 114, 114, 13 133, 3, 152, 17 1711 and 190 give pr od ucts uc ts m ad e u p of a series of nines with two varying digits and a zero or zeros. The Magical Number

76923 This Th is is one of the th e most curious n umb ers in ari thm eti c. Look what wh at happ ha ppen en s when whe n you multip mul tiply ly this nu mb er by 1, 10,. 9, 12, 3 and 4 76923 76923 76923 76923 76923 76923

X 1 = 076923 X 10 = 769230 X 9 = 692307 69230 7 X 12 = 923076 9230 76 X 3 = 230769 X 4 = 307692

You get the same sequence of   d i g i t s  when  r e a d f r o m  top to bottom or from left to right. Th at 's not all. Mult iply the same num ber   by 2, 7, 5, 11,6 and 8 and see what happens! 76 92 3 X

2 =

15 384 6

76 92 3 X

7 =

53 84 61

76923 x

5 =• 38 46 15

76 92 3 X

11 11 =

84 61 53

76923 x

6 =

461 53 8

76 92 3 X

8 =

61 53 84

Some Special Numbers

81

You get anoth an oth er sequence of di gits th at reads the sam e from top to bottom and from left to right. It is notewo not eworth rthyy th at in each of the th e above two tables, table s, all the answers consis consistt of th e same sam e digits ar ra ng ed in dif ferent fer ent groupings and the sum of the digits in all the answers is 27. What a strange peculiarity this number has!  Number 37037

Th is is anoth an oth er curious nu mb er . Look at the following follo wing table: 370 37 37037 37037 37037 3703 7 37037 37037 37037 37037 37037 37037

X X X X X X X X X

3 = 6 = 9 *= 12 12 = 15 = 18 = 21 = 24 = 27 =

111111 222222 222 222 3333 33 3333 33 4444 44 4444 44 555555 555 555 6666 66 6666 66 777777 7777 77 888888 888 888 999999 999 999

 A Curious Multiplication—Addition

Here is a series of multiplications—additions that bring out the nine digits in the natural and also in their inverted order: (I (12 (123 (1234 (12 34 (12345 (123456 (123 456 (1234567 (1234 567 (12345678 (123456789 (1234 56789

X  8) + 1 = 9 X 8) + 2 = 98 X  8) + 3 = 98 9877

x X x x x X

8) 8) 8) 8) 8) 8)

+ + + + + +

4 5 6 7 8 9

= = = = = =

98 76 9876 98765 9876 5 987654 987 654 9876543 9876 543 98765432 9876 5432 98765432 9876 543211

2

Figuring Made Easy l

Yet another: (1 X 9 ) + (12 X 9 ) +

2

=

11

3

=

111

(123 X 9 ) +

4

=

1111

(1234 X 9) +

5

=

11111

(12345 X 9) +

6

=

111111

(123456 X 9) +

7

=

1111111

(1234567 X 9) +

8

=

11111111

(12345678 X 9) +

9

(123456789 X 9 ) +

9

111111111 =

1111111111

The Unique Number

He re is is the th e smallest nu mb er , of which whi ch the al te rn at e figu figure re?? are zeros, and which is divisible by nine and also by 11. 90

90

90

90

90

90

90

90

90

90

9

Dividing it by 9, we get: 101010101010101010101

Dividing it by 11 gives: 82644628099173553719 This Th is again is a curious curio us nu mb er in th at it con tai ns various sequences of identi id entical cal nu mb er s in dir ect an d reversed orders order s  —826  —8 2644 a n d 46 4628 28,, 17355 a n 4 553 55371, 71, 1735 a n d 53 5371 71 a n d p erer haps various others. The Versatile Number g

Here is one more interesting characteristic of number 9. Multiply 9 by 21. We get the product 189. Mult Mu ltipl ipl y 9 by 32 321. 1. We get the pro du ct 2889. 2889. If you carry on successive multiplications in this order with inverted digits, one being bei ng add ed to the row each time, time , the th e successive successive multi pli  p liee r s will wi ll b e 21, 32 321, 1, 43 4321 21,, a n d so on u p t o t h e f u l l row ro w of

Some Special Numbers

83

digits dig its inve rted, 987654321. 987654 321. Let us now arrange the successive multiplications in the pyramidal form: 21 X 9 = 321 X 9 = 4321 X 9 = 54 32 1 X 9 = 654321 X 9 = 7654321 X 9 = 87654321 X 9 = 987654321 X 9 = 10987654321 X 9 =

18 9 28 89 38 88 9 48 88 89 58 88 88 9 68 88 88 89 78 888 88 89 88 888 888 89 988 888 888 89

7

GAMES TO PLAY WITH  NUMBERS

To Find tht Age of a Person

(1)) Ask (1 Ask your fr ie nd to multiply multi ply his h is age by 3 and an d add ad d six six t o the th e pro produc duc t, then th en ask him to divide the th e last nu numb mber er by thre e and tell you th e result. Subtr Su btr act ac t two from tha t result and give him the number. That will be his age. (2) You can also also tell the th e age of a person perso n who is und u nder er 21, by by the help of the following table: A

B

C

D

E

1 3 5 7 9 11 13 15 17 19

2 3 6 7 10 11 14 15 18 19

4 5 6

8 9 10 11 12 13 14 15

16 17 18 19 20

'/

12 13 14 15 20

Ask the person to tell you in which columns his age occurs and then the n ad d toge ther the r the numbe num bers rs at the top of those columns and the sum will be his age.

Games to Play with Numbers

85

Fo r exam ex ampl ple, e, if he says th at his age is fou found nd in columns colu mns C an d E, t h e n 4 + 16 = 20 is his age . (3)   A Nice Party Trick

If you wish to impress your friends with a trick of predicting a nu mb er , writ e ou t th e nu mb er 37 on a piece of  pa  p a p e r a n d k e e p it in yo your ur p o ck et. et . Ask o n e of y ou ourr f r i e n d s to call out any an y nu mb er fo form rmed ed of th re e ident id entical ical figu figure res. s. Ask Ask hi m to ad d u p these th re e figu figure ress and an d divide divi de th e nu mb er by the th e answer. T h e n you take out the th e piece of  pa  p a p e r f r o m y o u r po pock cket et on w h i c h is w r i t t e n n u m b e r 37! Wa tc h the reaction on everybody's face! The y'l l be amazed. You see, whate wh ate ver ve r mi ght be th e nu mb er of the th re e identical figures, the result is always the same 555 divided by 5 + 5 + 5 gives 37. 37. 3333 divided by 3 + 3 + 3 give 33 givess 37. 37. 888 divided by 8 + 8 + 8 gives gives 37 37.. (4) (4 ) Another Party Trick T o play pl ay this trick, tri ck, you need at least fou fourr othe ot herr persons  besi  be side dess yo youu in t h e p a r t y . Ask your neighbour to write down a three-digit number— any three-d thre e-digit igit nu mber mb er , withou wit hou t reservations. And ask ask hi m to write wri te th e same nu mb er alongside, along side, which gives a six-digit number. Then ask him to pass the slip to his neighbour, whom you will ask to divide the six-digit number by seven. Do n' t wor ry abo ut the nu mb er not dividing divid ing exactly by seven-—it will.  No  N o w ask h i m t o pass pa ss t h e r e su lt to his hi s n e i g h b o u r , w h o m you will ask to divide the number by 11. Again, Aga in, do n' t worry abo ut the nu mb er not dividing dividi ng by 11  — it will wi ll.. Ask the th e result resu lt to be passed again aga in to his neighb nei ghbour our,, whom who m you ask to divide by 13. It will divide by 13—don't worry.

86

Figuring Made Easy l

 Now  No w ask t h a t per pe r son so n to give gi ve yo youu b a c k t h e slip sl ip,, t h e p a p e r well well folded folde d so t h a t you don't don 't see the th e result res ult,, an d pass pass th at slip on, with wi thou outt un unfol foldi ding ng it to you yourr neig ne ighb hbou ourr who firs firstt wrote down the number, at the start. I t will will be th e same sam e he wrote on the slip of pa p er to begi be ginn with! Surprised? Here's the secret. Let's assume th at the firs firstt nu mb er th at was wri tte n was the number 498. A similar num b er wri tte n alongside it would mak e i t 498498. In actual fact what has happened to the number is that it has been multiplied by 1000 and then the original number has been added to it. 498 x 1000 + 498 = 498498 4984 98 which wh ich is th e same sa me as

498 X 10 1000 0011

which equals 498498 And 1001 = 7 X 11 11 X 13 Therefore what has actually been done is, the number was firs firstt multip mul tip lied by 1001 1001 and then gradu ally, all y, han d to han d divided by 1001! (5)) Ask your frien (5 fr ien d to write wri te a multi-digit multi- digit num n um be r — bu t with one condition—the number should not end with a zero. Ask Ask him to to ad d up the th e digits and subt su bt ract ra ct the tot al from the original origi nal nu mber mb er.. Th en ask ask him hi m to cross cross out anv one of the digits and an d tell you the rem ain ing numb ers. You do n' t need to know the th e original nu mb er nor wha w hatt he ha d done with it. You can 'p op ' tell him the exact nu mb er he ha d cross crossed ed out. How?

Games to Play with Numbers

87

Here's how you do it. Very Ve ry simple sim ple!! All you have ha ve to do is to find find th e digit which whi ch adde ad dedd to th e num n umber ber s he gave you will for form m the nearest near est number divisible by 9. And that will be the missing number. For exampl exa mple, e, if he had in mi nd the th e nu mber mb er 9241, 9241, he ad ds up the four digits and subtracts the total from the original number. He gets 9241 — 16 = 9225. Supposing Supp osing he crosses out th e 5 an d gives gives you 9, 2 an d 2, add ing in g 9, 2 an d 2 you yo u get 13, and the nearest number to 13 which should not be less less t h a n 13 and an d is divisible divis ible by 9 is 18. Th eref er ef or e, yo youu give the th e missing nu mb er as 18 — 13 = 5. (6) Guessing the Birthday

Wou ld you like to surpri sur prise se your fri end en d by guessing his  bi  b i r t h d a y ? You Yo u can ca n t ell el l y o u r f r i e n d t h e m o n t h a n d t h e date of his birth very easily. Here is how to do it. First of all, ask your frie fr iend nd to keep kee p in mind two n umbe um be rs, rs , the th e nu mb er of the mont mo nthh in which he was bor n an d th e nu mb er of the th e da te of the th e mon th. Th e month s, of course, are num ber ed from 1 to 12, begin ning from J an ua ry to Dece mber. mbe r. Th en you ask him hi m to multiply mult iply the nu mb er of the th e mo nt h by 5, add to this th is 6 an d mul tip ly it by 4 an d again aga in add ad d 9. Once On ce again agai n ask him to mult iply the num n um be r  by 5 a n d a d d t o it t h e n u m b e r of t h e d a t e on w h i c h h e was born. Wh en he fini finish shes es the th e calcu ca lcula latio tions ns,, ask ask him hi m lor th e fina finall result. T h en men tally tal ly you sub s ubtra tra ct 165 fro m the th e fina finall result. Afte r the subtra ction, th e rema inin g num ber gives gives the answer. The last two digits of the number give you the date of the mo nt h and an d the th e firs firstt digit digi t or th e first two digits give you the number of the month. For Fo r exam ex ampl ple, e, if you yourr fr ie nd 's fina finall result resu lt is 12 1269 69,, when wh en you subt su btra ract ct 16 165, 5, you get 110 1104. 4. From Fr om this nu mb er you know that he was born on November 4th. I am sure you would like to know the trick behind it. T h e direct di rection ionss you you give to your frien fr ien d are, a disguised disguised way wa y

88  Figuring Made Easy l

of add ing in g 16 1655 to the th e nu mb er ooff the month mon th , multipl ied by / 100. 10 0. Wh en the nu mb er 165 is taken take n away fr om the th e total, tot al, the number of the day and one hundred times the number of the month are left. Ask Ask yoifr frie fr iend nd s to pro prove ve th at 7 is equal eq ual to one-ha one -half lf of 12.. The y will natu 12 na tural ral ly give up. T h e n wr ite it e twelve in Roman numerals, and then draw a horizontal line through the middle, thus: VH~ AII Th u s you can see th at seven is one-ha one -half lf of twelve, the th e upper half being VII. Ask you yourr frie fr iend ndss to add ad d five an d six six in such a way that th at the th e sum will be nine . Th ey will will give up. T h e n make ma ke the following six strokes 111111. And to them add the following five strokes thus: I I I I I I

l \ ! l l \ l IE

Ask Ask your friend fri end s to pu putt down a three-digi three -digitt nu mb er un un-known to you. T ell el l th em to reverse t he digits and to sub tra ct th e smaller nu mb er from the grea ter. Th en ask ask th e m to give t h e firs firstt digit of th e result, res ult, whe reup re upon on you will be able ab le to t o give the t he e ntir nt iree answer, ans wer, by subtract subt racting ing th e firs firstt digit given to you from fr om nin e, whic wh ichh will fo form rm th e last digit an d, of course, the th e mi dd le dig it will always  be  b e n ine. in e. For exam ple, if you r friends fri ends have in mind mi nd th e nu mber mb er 843, the y will r everse ever se it it an d get 348 348.. Fr om 843 they th ey will will ta ke ou outt 348, give yo youu 4 as th e firs first, t, digit dig it of th e answe r.  Now  No w t h a t yo youu g o t t h e first d i g i t , a ll t h a t yo youu h a v e t o d o is subt su btra ract ct 4 fr om 9 and an d get 5 as the last digit di git . An d, of course, the th e mi dd le digit will always always be nine. nin e. Ther Th eref ef or e you give the answer as 495. Th is applies appl ies only when the digits digit s in the units uni ts an d hu hundndreds places are not equal.

7

 PUZZLES TO PUZZLE YOU

Her e ar e a few puzzles to tickle yo ur imaginatio imagi nationn and to excite your interest in f urt her arithme ari thme tica l inquiri inquiries. es. Thes e  puzzl  puz zles es a r e n o t r i dd ddle less m a d e t o dece de ceiv ivee yo youu o r j u s t pu puzz zzle less without witho ut purpose purpos e made ma de only to tease, bu t they are straightstraigh tforward forw ard exercises in reason that th at could cou ld give you many man y hou hours rs of fun and pleasure. (1) The Gong

Supposin Sup posingg a clock takes 7 seconds to strike 7, 7, how long will it take for the same clock to strike 10? (2)) A Problem of  Candy (2  Candy If 6 men me n can pack 6 packet pac ketss of candy ca ndy in 6 minutes, minut es, how many men are required to pack 60 packets in 60 minutes? (3) A Question of Distance It was a beaut ifu l sunny sunny morning. T he air ai r was was fresh an d a mild mi ld wind was blowing agains aga instt my windscr wind screen een.. I was driv dr ivin ingg from fr om Bangalore to Brindav Brin davan an gardens. garden s. It took me 1 hour and 30 minutes to complete the journey. After Afte r lun ch, ch , I retu re turn rned ed to Bangalore. Bangalo re. I drove for 90 minutes. How do you explain it?

90

Figuring Made Easy l

(4) Can Ca n you writ wr it e 23 with wi th only 2's, 45 only onl y with wi th 4's and 1000 with only 9's? (5) Do you notice anything interesting in the following multiplication? 138 X 42 = 57 5796 96 You You will not e th at ther th eree are nine nin e digits in the mult ipli cation cati on and an d all t he nine digits a re d ifferent iffe rent.. Can you think up other similar numbers? (6)

Three Fat Men

In my neighb nei ghb our ourhoo hoo d lives lives a very fat m a n who weighs 200 kilos. H e has ha s two very ver y fa t sons who wh o weigh wei gh 100 100 kilos each. O n a festival day , the y deci ded to go across across th e river riv er on o n a bo boat at to visit some rel ations ati ons.. But th e bo at could carry a maximum load of only 200 kilos. Can you tell tel l how th ey mana ma nage gedd to get across the t he river  by  b y b oat? oa t?  Gooseberries (7) A Problem of  Gooseberries

Wh en l was a lit tle giri, one day my mo th er ha d left a bowl b owl of gooseberr goos eberries ies to t o be sha s hare redd betw be twee eenn my two sisters La li th a, Va sa nt ha and an d myself. I went ho home me firs first. t. I at e what wh at I th ou gh t was my share sha re of gooseberries and left. T h en Lali La lith thaa arrived arri ved.. She thou th ought ght she was the firs firstt one to arrive arr ive an d ate the nu mb er of gooseberr gooseberries ies she thought was her share and left. Lastly, Vasantha arrived. She again aga in th ou ough gh t she was the firs firstt on e to arrive arr ive an d she took wh at she thou th ough ghtt was he r sha s hare re an d left lef t 8 gooseberries in the bowl. Wh en we th re e sisters sisters met me t in the th e evening, even ing, we realised what ha d hap pen ed and my mo th er distributed the remaining 8 gooseberries in a fair share. How did my mother do it? (8)) Can (8 Ca n you writ wr itee 1, by using usi ng all al l th e ten te n digits?

Puzzles to Puzzle You

91

 Mystery of the Missing Paisa

Tw o women were selling marb ma rbles les in th e market mar ket pl ac e  — one at thr ee a paisa and an d th e othe r at two a paisa. O n e da y they bo both th were obliged to re tu rn home when each had thir ty marbl es unsold. Th ey put together the two lots of marb ma rb les le s and hand ha nd in g them the m over to a friend f riend asked he r to sell them th em at five five for 2 paise. Acco rding rdi ng to th ei r calcul cal cul ation, ati on, af te r all, 3 for on e paisa and an d two for on e  pa  p a i sa was wa s exac ex actl tlyy t h e same sa me as 5 f o r 2 pais pa ise. e. But when wh en th e takings were hand ha nd ed over to them, th em, th ey were both most surprised, because the entire lot together h ad fetc fe tche he d only 24 paise! If, If , however, howev er, they ha d sold thei th eirr marb les sepa rately, rate ly, the y would have fetched fetc hed 25  pai  p aise se..  No  N o w wh e r e d i d t h e o ne p aisa ai sa go? Can you explain the mystery? The Pen and The Pencil

I bo ug ught ht a pen pe n an d a pencil pen cil for Rs 1.10 1.10 p. If th e pe n costs costs 1 ru pee pe e more mor e th an th e pencil, penc il, how much mu ch does t h e  pe  p e n c i l cost? cos t? Can you write 1000 by using eight 8's? There are many possibilities. Her e is an epi tap h of the celeb rated Greek math ema tician of 250 AD, Dio ph phant ant us. Ca n you calcu late his ag e from this? Dio ph phant ant us passed one-sixth one-sixth of his life in childhood, childh ood, one twelf th in youth, and one-seventh more as a bac hel or; five five years af te r his marr ma rr iage ia ge , a son was bo rn who died die d four fo ur years before his f at h er at half his fin final al age. How Ho w old ol d is Diophantus? I hav e a little friend fri end name na me d Raji Ra ji v. He bou bought ght a used cricket ball ba ll for 60 paise only. Hav ing boug b oug ht it, i t, he somehow some how di d not fin findd it to his liking likin g and an d so when wh en a fr ie nd of his offered of fered h im 70 paise, p aise, he sold sold it to h i m .

92

Figuring Made Easy l

However, Howe ver, hav ing sold sold it, Raj iv felt bad and decide d to  bu  b u y it b a ck f r o m h i s f r i e n d by o f f e r i n g hi h i m 80 pais pa ise. e. Having Hav ing bou bought ght it, it , once again Raj R aj iv felt th at he did not really real ly like th e ball ba ll,, and an d so he sold it again ag ain fo forr 90 paise pais e only. Did Raji Ra ji v mak e at all any profit in this tran sacti on? If so, how much? (14) Can you write 30 using any other 3 identical digits except 5's? (15) Ta ke the th e eight eigh t digits 1, 2, 3, 4, 5, 6, 7 an d 8, With Wi thou outt the aid of any othe ot he r digits and using th e additio add itio n and an d subtraction signs only, can you make 100? (16) At this mo me nt , it is 9 p.m. p. m. Can you tell me wh at time ti me it will be 23999999992 hours later?