3cs of macubDepartment of mathematics educationSouth Travancore Hindu College of Education proudly presents

Macub magazine mathematicseducation department *STHE* Nagercoil Kumarimacub magazine2014Our Principal&PatronDr.B KRISHNAPRASAD

Between us..This magazine of South Travancore Hindu College of Educations Mathematics departments presentation, we named it as 3cs . The idea of publishing a magazine on behalf of the department of Mathematics Education, the things came in mind is, it should be contain some valuable information about the facts and ideas of basic mathematics in day to day life and another is, to show, at least express or direct a dot or a pin-point size of the beauty of Mathematics to its reader. For example, more or less, even we are theoretically studying mathematics in a good extent, the practical knowledge of the theory or practicing the theory , where does we apply it? and what is the use of this? Do we study this for merely to score marks in exam? etc., are the great questions came in the mind of a common mathematics student-Scholar.Traditionally, we use some scales to measure the matters in our daily life. But now the scale to measure the same are changed to international unit, like; How many Millies makes one ounce? How long is ayard? To answer such questions not entirely, but we try to collect such information with in our limit; in this magazine.We call this magazine as 3cs means Creations, Collections and Copies includes our student teachers creations, collections from various sources and copies of some extracts.We the editorial board, use this opportunity to thank the management of Our College and Our Principal for their support and permission given us to publish this magazine. Our gratitude also to all the Staff members of our College for their support and suggestions.We are thankful to all the student teachers for their contribution to this magazine.Nagercoil. Chief Editor. eP kdpjdhfg; gpwe;jpUf;fpwha; eP tho;e;J kiwe;jjw;Fg;gpd;dhy; Xh; mopahj mwpFwp vijahtJ tpl;Lr; nry; Rthkp tpNtfhde;jh;.

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Our source of inspiration

Fun with your bdayIsnt your bday?Fun with bday1.4 x your birth date2.+ 133.x 254. 2005.+ month you are born6.x 27. 408.x 509.+ last 2 digits of year of birth10. 10500

Magical number with your age259(your age)39What do you find?Isnt your bday?

Fun with your ageBirth year (last 2 digits) + your age = ------What do you find?

Mathematics is the door and key to the sciences.- Roger BaconCan you read this numeral?7, 346, 648, 004, 560, 986, 215, 348, 444, 286, 445, 305, 113, 039, 140, 046, 960, 678, 582, 256, 003Now read this----------Seven Vigintillion,Three hundred forty six Novemdecillion,Six hundred forty eight Octodecillion,Four Septedecillion,Five hundred sixty Sexdecillion,Nine hundred eighty six Quindecillion,Two hundred fifteen Quattourdecillion,Three hundred forty eight Duodecillion,Four hundred forty five Nonollion,Three hundred five Otollion,One hundred thirteen Septillion,Thirty nine Sextillion,

One hundred forty Quintillion, Forty six Quadrillion,Nine hundred sixty Trillin,Six hundred seventy eight Billion,Five humdred eighty two Million,Two hundred fifty six thousand and three.The essence of mathematics is its freedom.-George Cantor.Letter to Mathematicscosec 14, Alpha flats,Beta Street, Hyperbola district.My dear tan , Well. How are you? How is your angle of elevation and angle of depression? How are your parents sin and cos? I heard that your mom sin is suffering from trigonometric fever. Ask her consult doctor Ramanujan. He is a specialist in arithmetic operation and fractional surgery. Convey my love to all especially to your brothers sec , cot . Ask your mom to subtract sorrow, multiply joy and divide it among yourself. No more to add.

Yours lovingly,Cosec .Address on the envelopeMr. tan ,26, z complex, parabola district.

Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician-friedrigh ludwigRamanujam NumbersSome numbers can be expressed as sum of cubes of two numbers in two different ways. These peculiar numbers are called Ramanujam numbers. These are given below:- 1729 = 123 + 13 = + 20683 = + = + 39312 = + = 153 + 40033 = + = + 171288 = + = + 195841+ 57 3 = 93 + 583 515375 = 153 + 803 = 543 + 713Fun with 37111 / (1 + 1+ 1) = 37222 / (2 + 2 + 2) = 37333 / (3 + 3 + 3) = 37444 / (4 + 4 + 4) = 37555 / (5 + 5 + 5) = 37666 / (6 + 6 + 6) = 37777/ (7 + 7 +7) = 37888 /(8 + 8+8) = 37999/(9 + 9 +9) = 37Mathematics is the science of what is clear by itself - carl gustav jacobiPlaying with numbers3cs of macub

Department of Mathematics, South Travancore Hindu College of Education, Nagercoil.Expressing numbers 1 to 10 using five 2s only. 1 = 2 + 2 2 - 2/2. 2 = 2 + 2 + 2 - 2 - 2.3 = 2 + 2 2 + 2/2.4 = 2 x 2 x 2 - 2 2.5 = 2 + 2 + 2 - 2/2.6 = 2 + 2 + 2 + 2 2.7 = 222 - 2 2 2.8 = 2 x 2 x 2 + 2 - 2.9 = 2 x 2 x 2 + 2/2.10 = 2 + 2 + 2 + 2 + 2 .Expressing numbers 1 to 10 using 3s only.1 = (3 3) x (3 3)2 = (3 3) + (3 3)3 = (3 + 3 + 3) 34 = 3/0.3 - (3 + 3)5 = (3 + 3) - ( 3 3)6 = 3 + 3 + 3 37 = 3 + 3 + 3/38 = (33 3) - 39 = (3 x 3) + 3 - 310 = (33 - 3) 3.

There is no royal road to geometry-EuclidExpressing numbers 1 to 10 using four 4s onlynoexpressionnoexpression

144 446 + + -

24/4 + 4/4

74/4 + 4 +

34/4 + 4 -

8 + + +

44 + 4 - -

94 + 4 + 4/4

54/4 + +

104 + + +

More or lessInfinite people shouting to be heard equalsgreater music that adds and subtractsfrom the integral event of exposing yourselfto simplify.Look to introduce yourself, make the most.Divide the room up and people multiply.Young actors gather within parentheseshoping to gain more or less dream

He who can properly define and divide is to be considered a God -PlatoHighest one digit to eight digit prime numbersNumber of digitHighest prime

17

297

3997

49973

59991

6999983

79999991

8999999989

A special Number 381654729.The 9 digit number 381654729 was all the digits from 1 to 9 exactly once. Starting from the left, the first digit is divisible by 1, the first 2 digits are divisible by 2, the first 3 digits are divisible by 3, the first 4 digits are divisible by 4, and so on till all the 9 digits together are divisible by 9. There is only one suchnumber381654729.I am a special number.Me contains all the digits from 1 to 9.God does not care about our mathematical difficulties. He integrates empirically - Albert Einstein666, The number of the BeastThe number 666 is the number of the beast . It is also called the sign of devil. The number 666, being the number of the beast is a symbol for a person or organization which persecutes the people of God. It is also interesting because of its many numerical properties. Here is a com-pendium of mathematical facts about the number 666.i) The Roman numerals are:I V X L C D M1 5 10 50 100 500 1000Excluding M=1000, DCLXVI = 500 + 100 + 50 + 10 + 5 + 1 = 666ii) A prime number is a natural number that has exactly two distinct divisors. 1 and itself. The first seven prime numbers are 2, 3, 5, 7, 11, 13, 17. The beast number is equal to the sum of the squares of the first primes. + + + + + + = 666iii) The number 666 is a sum and difference of the first powers. 666 = - + .iv) There are exactly two ways to insert + signs into the sequence 1 2 3 4 5 6 7 8 9 to make the sum 666 and exactly one way for the sequence 9 8 7 6 5 4 3 2 1.666 = 1 + 2 + 3 + 4 + 567 + 89666 = 123 + 456 + 78 + 9666 = 9 + 87 + 6 + 543 + 21.v) 666 = + + + 6 + 6 + 6.vi) 666 is the divisor of 1 2 3 4 5 6 7 8 9 + 9 8 7 6 5 4 3 2 1

I see it, but I don't believe it.-George Cantor*Happy numbers *139 is a Happy Number. To find out whether a Number is a happy or not square its digits add them up and go on doing this. If you get 1 in the end, the given number is a happy number. With 139, we reach 1 in the following 5 steps. + = 1+9+81 = 91 = 81 + 1 = 82 = 64 + 4 = 68 = 36 + 64 = 100 = 1 The magic number 91 9 = 09 = 0+9 = 92 9 = 18 = 1+8 = 93 9 = 27 = 2+7 = 9 4 9 = 36 = 3+6 = 95 9 = 45 = 4+5 = 96 9 = 54 = 5+ 4 = 97 9 = 63 = 6+3 = 98 9 = 72 = 7+2 = 9 9 9 = 81 = 8+1 = 910 9 = 90 = 9+0 = 9 This is true for all multiples of 9.Mathematics is the queen of the sciences and number theory is the queen of mathematics.-Gauss

Rule of maths if it seems easy,You are doing it wrong.This is MATHEMATICS nuMber Algebra daTa cHance gEometry Measurement cAlculus ariThmeticstatIcsSpaCe StatisticsMagic numbers 11 11 = 121111 111 = 123211111 1111 = 1234321 11111 11111 = 123454321111111 1111111 = 123456543211111111 11111111 = 123456765432111111111 11111111 = 123456787654321111111111 111111111 = 12345678987654321In mathematics the art of proposing a question must be held of higher value than solving it - George CantorAbout PolygonA closed plane figure made up of several line segments that are joined together. The sides do not cross each other. Exactly two sides meet at every vertex.Types of Polygons 1. Regular Polygon2. Convex Polygon3. Concave PolygonRegular - Regular polygons are both equiangular and equilateral.Equiangular - all angles are equal.Equilateral - all sides are the same length.Convex - a straight line drawn through a convex polygon crosses at most two sides. Every interior angle is less than 180.Concave - you can draw at least one straight line through a concave polygon that crosses more than twosides. At least one interior angle ismore than 180.Area of a regular polygon = (1/2) N sin(360/N) Sum of the interior angles of a polygon = (N 2) 180.The number of diagonals in a polygon = N(N -3)The number of triangles (when you draw all the diagonals from one vertex) in a polygon = (N 2)

It is not enough to have a good mind. The main thing is to use it well. Rene Descartes.Generally accepted names of polygonsSides Namen N-gon3 Triangle4 Quadrilateral5 Pentagon6 Hexagon7 Heptagon8 Octagon9 Nonagon, Enneagon 10 Decogon11 Undecagon, Hendecagon12 Dodecagon13 Tridecagon, Triskaidecagon14 Tetradecagon, Tetrakaiecagon15 Pentadecagon, Pentakaidecagon 16 Hexadecagon, Dexakaidecagon17 Heptadecagon, Heptakaidecagon18 Octadecagon, Octakaidecagon19 Enneadecagon, Enneakaidecagon 20 Icosagon30 Tricontagon40 Tetracontagon50 Pentacontagon60 HexacontagonIt is not certain that everything is uncertain. Blaise Pascal70 Heptacontagon80 Octacontagon90 Enneacontagon 100 Hectogon, Hecatontagon1,000 Chiliagon

To construct a name, combine the prefix + suffixSides Prefix Sides Suffix20 Icosikai.. +1 ...henagon30 Triacontakai. +2 digon40 Tetracontalai + 3 . ..tigon50 Pentacontakai +4 ...tetragon.60 Hexacontakai. +5 pentagon70 Heptacontakai.. +6 . ..hexagon80 Octscontsksi.. +7 heptagon90 Enneacontakai +8 octagon +9 ennegon 10,000 Myriagon

Your answer is 10998(1) On the blank page write any five digit number. Now rewrite the number with and last digit intechanged.(2) Substract the smaller number from the greater one and write the result on another part of the paper.(3) Rewrite this number with and last digit intechanged directly under it.(4) This time add the two number and your answer is 10998.

God created the natural number, and all the rest is the work of man. Leopold Kronecker.magic numbers 37 3 = 11137 6 = 22237 9 = 333

37 12 = 44437 15 = 55537 18 = 66637 21 = 77737 24 = 88837 27 = 999Fantastic prime numbers with 3 31 331 3331 33331 333331 3333331 33333331these numbers are prime numbers. But the next number in the series is not a prime number. 19607843 17 = 333333331What we know is not much. What we not know is immense. Laplace.Conversion formulaeTo change ToMultiply

acreshectares0.4047

acresSquare feet43,560

acresSquare miles0. 001562

Atmosphere

Cms. of mercury76

Btu/hourHorse power0.0003930

BtuKilowatt-hour0.0002931

To change To Multiply

1 Centimetersfeet0.0328

2 centimetersinches0.3937

3 Feetcentimeters30.4801

4 Metersfeet3.2808

5 Metersinches39.37

6 Metersyards1.0936

7 Miles kilometers1.6093

8 kilograms pounds2.2046

9 Kilometersfeet3,280.833

10 Kilometersmiles0.6214

11 Kilometers/hour Feet/minute54.68

12 Kilometers/hourMiles/hour0.6214

13 Meter-kilogramsFoot-pounds7.2307

14 Meters/minuteCentimeters/second1.667

15 Meters/minuteFeet/second0.0547

16 KnotsKiometers/hour1.8532

17 Miles/hourknots0.8684

18 Miles/hourKilometers/hour1.6093

19 Miles/hourMeters/second0.447

20 Sq.metersSquare feet10.7639

21 Sq. metersSquare yards1.196

22 Sq.miles Square kilometers2.59

23 Sq. yads Square meters0.8361

24 Yardscentimeters91.44

25 Yardsmeters0.9144

26 Feet/minutesCent./seconds0.507

27 Foot-poundsMeter-kilometers0.1383

28 Gallonsliters3,785.4

29 gallonsliters3.7853

30 Gramsounces0.0353

31 GramsPounds0.0022

32 Inchescentimeters2.54

33 Inches Feet0.0833

34 LitersQuarts1.0567

35 OuncesGrams28.3495

36 Ounceskilograms208349x10(2)

37 PoundsGrams453.5924

38 QuartsLiters0.946

39 CentimetersInches0.3937

40 CentimetersFeet0.03281

41 Cubic feetCubic meters0.0283

42 Cubic metersCubic feet35.3145

43 Degrees Radians0.1745

44 DynesGrams0.00102

45 Fathoms Feet6.0

46 FeetMeters0.3048

47 FeetMiles(nautical)0.0001645

48 Feet/secondmiles/hour0.6818

49 FurlongsFeet660

50 FurlongsMiles0.125

51 Gallons(U.S.)Litres3.7853

52 GrainsGrams0.0648

53 GramsGrain15.4324

54 GramsOunces(avdp)0.0353

55 GramsPounds0.002205

56 Hectares Acres2.4710

57 HorsepowerWatts745.7

58 HorsepowerBtu/hour2,547

59 Inchescentimeters2.5400

60 LitersGallons(U.S.)0.2642

61 Literspecks0.1135

62 Miles feet5280

63 Miles(nautical)Miles(statute)1.1516

64 Miles/hourFeet/minute88

65 Ouncespounds0.0625

66 Radiansdegrees57.30

67 Fodsmeters5.029

68 Square feetSquare meters0.0929

Fun with your ageBirth year (last 2 digits) + your age = ------= 0.6 miles10 miles = 16.1 kilometers (km) 1 kilometer = 0.6 miles10 kilometers = 6.2 miles 0 Celsius = 32 Fahrenheit

1 mile = 1.6 kilometers (km) 1 kilometer = 0.6 miles10 miles = 16.1 kilometers (km) 1 kilometer = 0.6 miles10 kilometers = 6.2 miles 0 Celsius = 32 Fahrenheit

Mathematicians are born, not made. - Henry PoincareRAMANUJAMS TRUTH1 18 + 4 = 2212 18 + 6 = 222123 18 + 8 = 22221234 18 + 10 = 2222212345 18 + 12 = 222222123456 18 + 14 = 22222221234567 18 + 16 = 2222222212345678 18 + 18 = 222222222123456789 18 + 20 = 2222222222 Magic SquareAdd the number in any manner watch out for the answers which area always equal to 36947586980112233445

57687991122334446

67788102132435456

77718203142535566

61719304152636576

16272940516264755

26283950617274115

36384960717331425

37485970812132435

Wherever there is number, there is beauty- Diadochus Proclus.Where is the ONE RUPEE ?Three friends have a nice meal togeher and the bill is Rs.25.The three friends pay 10 each, which the waiter gives to the cashier.The cashier hands back 5 to the waiter. But the waiter cant split 5 three ways, so he gives thefriends one rupee each and keeps Rs.2 as a tip.They all paid Rs.10 & got Rs.1 back so eaach one paid 10-1=9.They were three = 3 x 9 = 27.If they paid Rs.27 &er kept 2.Totally 27 + 2 = 29.

Ha! Ha!! Ha!!!Maths teacher : If you have 12 chocolates and you give 5 to Aruna, 3 to Anitha and 4 to Kavitha then what will you get? LKG Terror : 3 New girl friends.Where did the other rupee go?

Facts do not speak. Henri Poincare.THE ANGEL NUMBER 421Rules Step 1 :Select any whole number. Step 2 :If it is an even number, divide by 2; if it is odd number multiply by 3 and add 1. Step 3 :Repeat the process mentioned in step 2 until you get the loop value 4, 2, 1 in repetition. Example Whole number is 15. 15 is an odd no; so (15 3) + 1 = 46 46 is an even no; so 46 / 2 = 23 23 is an odd no; so (23 3) + 1 = 70 70 is an even no; so 70 / 2 = 35 35 is an odd no; so (35 3) + 1 = 106 106 is an even no; so 106 / 2 = 53 53 is an odd no; so (53 3) + 1 = 160 160 is an even no; so 160 / 2 = 80 80 is an even no; so 80 / 2 = 40 40 is an even no; so 40 / 2 = 20 20 is an even no; so 20 / 2 = 10 10 is an even no; so 10 / 2 = 5 5 is an odd no; so (5 3) + 1 = 16 16 is an even no; so 16 / 2 = 8 8 is an even no; so 8 / 2 = 4 4 is an even no; so 4 / 2 = 2 2 is an even no; so 2 / 2 = 1 1 is an odd no; so (1 3) + 1 = 4 4 is an even no; so 4 / 2 = 2 2 is an even no; so 2 / 2 = 1 So the loop 4..2..1 goes on and on.Tip :The angel number 421is the smallest prime formed by the two powers in logical order from right to left.God created everything by number,weight and measure. Sir Issac Newton.Puzzles You can write one number on each face of the dice from 0 to 9 and you have to represent days from 1 to 31, for example for 1, one dice should show 0 and another should show 1, similarly for 31 one dice should show 3 and another should show 1.First dice : 0, 1, 2, 3, 4, 5second dice : 0, 1, 2, 6, 7, 86 can be used for both 6 and 9.jj;Jtk; tho;f;ifAk; fzpjKk; ey;ytw;iwf; $l;bf; nfhs; - jPatw;iwf; fop;j;Jf; nfhs; mwpitg; ngUf;fpf; nfhs; Neuj;ij tFj;Jf; nfhs; tsh;gpiw Nghy; tho;tpy; tsh;f nryitf; Fiwj;J tuitg; ngUf;Fmd;igg; ngUf;fp Mztj;ijf; Fiw gpwiu ek;gp thOk; tho;f;if epiyaw;wJ ey;ytUf;F ,izahf ,UThere are things which seem incredible to most men who have not studied Mathematics. Archimedes

Beauty of Maths Maths is beautiful

Step: 1 Any three digit number without repetition is chosen (for example - 916 is chosen).Step: 2 Reverse the number (ie., 619).Step: 3 Subtract the two numbers (ie., 916 619 = 297). The answer is 297.Step: 4 Reverse the above numbers (ie., 792)Step: 5 Add the two numbers (ie., 297+792 = 1089).Step : 6 Always, the answer will be 1089 It is not once nor twice but times without number that the same ideas make their appearance in the world. Aristotle.You told me about a golden rule

You told me about a golden ruleWhere life was divided into infinite possibilitiesBut somewhere along that divided lineI fell from graceAnd became only a remainderIn timeQuietly i count my mistakesHoping to shape a solution,To multiply the oddsAnd finish my suffering

Collected by jananiWhen you subtract from your lifeall the variablesthat don't add upnegatives cancels outthe hurt cancels outand it's not so bad to be emptysometimesBreaking down the finite walls around youin the quiet nightwhen you're still and silentI can feel the vibrationsI fall through the equationsand sink into theinfinitestarsin the sky

We cannot prove geometrical truths by arithmetics. Aristotle.our instructional meterials

One can always reason with reason. Heri Bergon.

Thought is a flash between two long nights, but this flash is everything.- Henri Poincareknow our shapes

Mathematics have created a universe from nothing. Heri Bergson.

Perfect numebers like perfect men are very rare. Rene descartes.

There is nothing strange in the circle being the origin of any and every marvel. Aristotle.

Everything existing in the universe is the fruit of chance and necessity. Democritus.

Geometry will draw the soul toward truth and create the spirit of philosophy. plato.The number you have thoughtStep1: Think of any number.Step2: Subtract the number you have thought with 1.Step3: Multiply the result with 3.Step4: Add 12 with the result.Step5: Divide the answer by 3.Step6: Add 5 with the answer.Step7: Take away the number you have thought from the answer, that is, subtract the answer from the number you have thought. *******

Tough MultiplicationStep1: If you have a large number to multiply and one of the numbers is even, you can easily subdivide to get to the answerStep2: Ex:32 x 125, is the same as: 16 x 250 is the same as: 8 x 500 is the same as: 4 x 1000 = 4,000

Only having one truth about each object, whoever finds it knows as much as can known about it. Rene descartes.NUMBER SYSTEMSINDIAN NUMBERS & MAYAN NUMBERS

We are servants rather than masters in mathematics.- Hermite.BABILONIAN NUMBERS

God ever arithmetizes. Carl Gustav JacobiRoman letters arent they beautiful

All the effects of nature are only mathematical results of a small number of aimmutable laws. - Laplace.Some numbers

The knowledge of which geometry aims is the knowledge of the eternal. Plato.Test of divisibility

DivisibilityConditionExample

Divisibility by 2

All even numbers (numbers having the d All even numbers (numbers having the digits 0, 2, 4, 6, 8, in the unit place All even numbers (numbers having the digits 0, 2, 4, 6, 8, in the unit place) are divisible by 2.) are divisible by 2.igits 0, 2, 4, 6, 8, in the unit place) are divisible by 2.For example, (1) 24 , 428, 3660, 73512 etc. are divisible by 2 (2) 91, 369, 4265, 8043, 2001 are not dpvisible by 2.

Divisibility by 3If the sum of the digits of a number is divisible by 3, then the number is divisible by 3. For example ( 1 ) 37236 is divisible by 3 as its sum of the digits 3+7+2+3+6 = 21 is divisible by 3. (2) 425372 is not divisible by 3 as its sum of the digits 4+2+5+3+7+2 = 23 is not divisible by 3.

Divisibility by 4 If the digits on tens and ones place are divisible by 4 or 0, the number is divisible by 4.i ) 6256 is divisible by 4 as 56 is divisible by 4. (ii) 3571900 is divisible by 4 as last two digit 00. (iii) 48230 is not divisible by 4 as 30 is not divisible by 4.

Divisibility by 5 If the digit on unit place is 0 or 5 the number is divisible by 5. ( i) 3214675 is divisible by 5 (ii) 135790 is divisible by 5(iii) 14213, 25551 are not divisible by 5.

Divisibility by 6If a number is divisible by 2 and 3 then it is divisible by 6(i)2318 is divisible by 6 as it is divisible by 2 and 3.(ii) 6530 is notdivisible by 6 as 6 + 5 + 3 +0 = 14 is not divisible by 3.

Divisibility by 7 The rule to check the divisibility of a number by 7 is complicated but explained as follows: Consider 11529602. Step (i) First group three-three digits from right as 11,529,602 first triplet of digits has positive sign, second triplet of digits negative, next positive and so on. Step (ii) Find the sum of these triplets +11-529+602 = 84 and 84 is divisible by 7, so the given number 11529602 is divisible by 7.149 884 826 149-884-826 = 91. 91 is divisible by 7 so 149884826 is divisible by 7.

Divisibility by 8The number formed by the last 3 digits is divisible by 8 or 000 then the number is divisible by 8.) 123480 is divisible by 8 as 480 is divisible by 8. (ii) 169324 is not divisible by 8 as 324 is not divisible by 8.(iii)423000 is not divisible by 8 as the last 3 digits are 000.

Divisibility by 9As in the case of divisibility by 3 if the sum of the digits is divisible by 9, then the numbe is divisible by 9.Example (1) 234531 is divisible by 9 as 2+3+4+5+3+1 = 18 is divisible by 9. Example (ii) 25617 is not divisible by 9 as 2+5+6+1+7 = 21 is not divisible by 9.

Divisibility by 10If the digit on ones place is 0 then it is divisible by 10. Example: 10, 0, 30, 110, 230, etc. are divisible by 10.

Divisibility by 11 If the sum of the digits at odd places in given number is either equal to the sum of the digits at even places in that number or differs from it by a number which is divisible by 11.Eample (i) consider 3748097sum of odd place digits : 3+4+0+7 = 14sum of even place digits: 7+8+9 = 24difference = 10 not multiple of 11 or 0. so it is not divisible by 11.Example (ii) consider 35937sum of odd place digits 3+9+7 = 19sum of even place digits 5+3 = 8difference = 11So the given number is divisibsle by 11.

Divisibility by 12 A number divisible by 3 and 4 then it is divisible by 12.

Divisibility by 13 As in the case of divisibility by 7. 149884826 Grouping 149 884 826sum +149-884+826 = 9191 is divisible by 13. So the number 149884826 is divisible by 13.

Divisibility by 14A number divisible by 2 and 7 is divisible by 14.

Divisibility by 15A number divisible by 3 and 5 is divisible by 15.

Divisibility by 16Divisible by 4 twice.

Divisibility by 17 Substract 5 times the last digit from the rest.and do as the example ifthe result is divisible by 17 the number is divisible by 17.Example: 221 : 22- (1 5 ) = 17 is divisible by 17.

Divisibility by 18Divisible by 2 and 9.

Divisibility by 19Add twice the last digit to the rest.and do as example ifthe result is divisible by 19the number is divisible by 19..

Example 437 43+(72) = 57 is divisible by 19.

Divisibility by 23Add 7 times the last digit to the rest. .and do as example ifthe result is divisible by 23the number is divisible by 23..3128 : 312 + (8 7) =368 368 23 =16. divisible by 23.

Divisibility by 27Sum the digits in blocks of three from right to left. If the result is divisibility by 27, then the number is divisible by 27. (ii) Substract 8 times the last digit from the rest. i) 2,644,272 : 2+644+272 = 918 = 27 34621 =62 (1 8) = 54 is divisible by 27.

Divisibility by 29Add thrice times the last digit to the rest. and do as example ifthe result is divisible by 29the number is divisible by 29.261 : 1 3 = 3; 3+26 = 29 is divisible by 29.

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HHH

me;jf; Fjpiu yhaj;jp;y; ve;jg; gf;fkhf ,Ue;J ghh;j;jhYk; %d;W miwfspYs;s Fjpiufs; fz;Zf;Fj; njupAk;. ,t;tpjk; 4 gf;fq;fspuy; ,Ue;J ghh;;;j;jhYk; xNu vz;zpf;if Fjpiufs;jhd; mtd; fz;Zf;Fj; njupAk;. ,jid jpde;NjhWk; khiyapy; kd;dd; cyh tUk; NghJ Fjpiufspd; vz;zpf;if rhp jhdh vd;;gij cWjp nra;J nfhs;thd;. nkhj;j Fjpiufisfg;gw;wp ftiyap;y;iy. xU ehs; jpBnud kio nga;jjhuy; xUtd; rpy Fjpiufis nfhz;L te;J murdJ Fjpiu yhaj;jpy; fl;Lf; nfhs;s mDkjp Nfl;lhd;. mg;nghOJ murd; mtdplk; ehd; ve;jg;gf;fk; ,Ue;J ghh;j;jhYk; xNu vz;zpf;if Fjpiufs; ,Uf;f Ntz;Lk;. mjw;Nfw;whw;Nghy; fl;;bf; nfhs; vd;W nrhd;dhd;. te;jtDk; mjd;gb fl;b tpl;lhd;. murd; tof;fk; Nghy; te;J ghh;j;jhd;. ve;jg; gf;fkpUe;J ghh;j;jhYk; xNu vz;zpf;if Fjpiufs; njhpe;jd. murd; jpUg;jpaile;jhd;. mLj;j ehs; kio epd;W tpl;ljhy; mtd; Fjpiufisf; nfhz;L nry;y mDkjp Nfl;lhd;. murDk; mDkjpaspj;jhd;. Mdhy;mtd; vt;tsT Fjpiufs; nfhz;L te;jhNdh mijtpl $Ljyhf xU klq;F Fjpiufisf; nfhz;L nrd;whd;. murd; mtdplk; eP nfhz;L te;j Fjpiuia tpl $Ljyhf xU klq;F Fjpiufis nfhz;L nry;fpwhNa Vd; vdf; Nfl;lhd;. mjw;F mtd; ePq;fs; Fjpiu yhaj;jpy; nrd;W ghUq;fs; cq;fs; fzf;Fg;gb ve;jg;gf;fk; ghh;j;jhYk; Fjpiufspd; vz;zpf;if mtdJ vz;zg;gb rhpahf ,Ue;jJ. murDk; Mr;rhpag;gl;lhd;. Fjpiufis nfhz;L te;jtd; vt;thW Fjpiufis yhaj;jpy; fl;bdhd;? vj;jid Fjpiufis fl;bdhd; vd;gij fz;LgpbAq;fs;.Perfect clarity would profit the intellect but damage the will. Blaise Pascaltpil:1. Kjypy; Fjpiu yhaj;jpy; murd; fl;bapUe;j Fjpiufs; tptuk;.444

44

444

2. ,d;ndhUtd; Fjpiufis nfhz;L te;;J fl;bagpd;G Fjpiufspd; tptuk;.

363

66

363

ve;;;;j %iyapy; epd;W ghh;j;jhYk; 12 Fjpiufs; njhpAk;. nkhj;j Fjpiufs; 36. mjhtJ 4 Fjpiufis mtd; nfhz;L te;J fl;bAs;shd;.3. mtd; Fjpiufis mtpo;j;Jf;nfhz;L nrd;wgpd; Fjpiufspd; tptuk;.

4. 525

22

525

ve;j %iyapy; ,Ue;J ghh;j;jhYk; 12 Fjpiufs; njhpAk;. nkhj;j Fjpiufs;;;;; 28. mtd; nfhz;L te;j Fjpiu 4I tpl xU klq;F mjhtJ 8 Fjpiufis mtpo;j;Jr; nrd;Ws;shd;.

Mathematics is the supreme judge; from its decisions there is no appeal. Tobias DantzigGjph; ???vilia Jy;ypakhf mstplf;$ba xU juhR cs;sJ. 8 fhrfs; xNu msthd viliaf; nfhz;lJ. ,d;Dk; 1 fhR 8 fhRfis tpl vilapy; rw;W FiwT. tbtikg;igf; nfhz;L vil Fiwe;;;;j fhrpidf; fz;lwpa KbahJ. juhrpid 2 Kiw kl;LNk gad;gLj;jp vil Fiwe;j fhrpidf; fz;lwpa Ntz;Lk;.tpil:Kjy; Kiw gad;gLj;Jjy;:-nkhj;jk; 9 fhRfs; cs;sd.

juhrpd; xU jl;by; 3 fhRk;> kw;nwhU jl;by; 3 fhrpidAk; ,l Ntz;Lk;.

juhrp;;y; eLepiyik fhl;bdhy; kPjKs;s 3 fhRfspy; jhd; vil Fiwe;j fhR ,Uf;f Ntz;Lk;. ,uz;lhtJ Kiw gad;gLj;jyhk;.

mt;thW eLepiy my;yhky; xU jl;L vil Fiwitf; fhl;bdhy; me;jj; jl;bYs;s %d;W fhRfspy; VNjh xd;W vil Fiwe;jJ.

jhurpid ,uz;lhtJ Kiw gad;gLj;jyhk;.

,uz;lhk; Kiw gad;gLj;Jjy;:

vil Fiwe;jJ vd;W ehk; fz;lwpe;j 3 fhRfis ifapy; vLj;Jf;nfhs;sTk;.

vil rkkhd 6 fhRfis jdpNa itj;J tplTk;.

ifapYs;s 3 fhrfspy; xU jl;by; xU fhrpidAk; kw;nwhU jl;by; xU fhrpidAk; itj;Jf; nfhs;s Ntz;Lk;.

juhR eL epiyia fhl;bdhy; ifapYs;s fhR jhd; vil Fiwe;jJ.

mJ my;yhJ juhrpy; vil Fiwe;j jl;L fhz;gpj;jhy; me;jj; jl;bYs;s fhR vil Fiwe;;;;;;jjJ.

Questions are creative acts of intelligence. Frank Kingdom.jkpo; ,yf;fKiwmNugpa ,yf;fk; nrhy;ypy;0 Ropak;9 njhz;L xd;gJ90 njhd;gJ1000 Mapuk;9000 njhs;;shapuk;10000 gj;jhapuk;100000 E}whapuk;1000000 gj;J E}whapuk;10000000 Nfhb100000000 mw;Gjk; (gj;J Nfhb)1000000000 epfh;Gjk; (E}W Nfhb)10000000000 Fk;gk; (Mapuk; Nfhb)100000000000 fzk; (gj;J Mapuk; Nfhb) 1000000000000 fw;gk; (E}W Mapuk; Nfhb)10000000000000 epfw;gk; (Mapuk; Mapuk; Nfhb)100000000000000 gJkk; ( Nfhb Nfhb)1000000000000000 rq;fk;10000000000000000 nty;yk;100000000000000000 md;dpak;1000000000000000000 mh;j;jk;10000000000000000000 guhh;j;jk;100000000000000000000 G+hpak;1000000000000000000000 Kf;NfhbTo learn you must want to be taught.

ePs tha;g;ghL

10 Nfhd; 1 Ez;zZ10 Ez;zZ 1 mZ8 mZ 1 fjph;j;Jfs;8 fjph;j;Jfs; 1 JRk;G8 JRk;G 1 kaph; Ezp8 kaph; Ezp 1 Ez;kzy;8 Ez;kzy; 1 rpWfLF8 rpWfLF 1 vs;8 vs; 1 ney; vz; $Wfspd; ngah;fs;

1 xd;W Kf;fhy; miu fhy; 1/5 ehYkh ehd;F 3/16 %d;W tPrk; Kk;khfhzp3/20 Kk;kh %d;Wkh1/8 miuf;fhy;1/10 ,Ukh

1/16 khfhzp tPrk;1/20 xUkh3/64 Kf;fhy; tPrk;3/80 Kf;fhzp1/32 miu tPrk;1/40 miu kh1/64 fhy; tPrk;1/80 fhzp1/160 miuf;fhzp1/320 Ke;jphp3/320 miuf;fhzp 1/57511466188800000000 Ez;kzy;8 ney; 1 tpuy;12 tpuy; 1 rhz;2 rhz; 1 Kok;4 Kok; 1 ghfk;600 ghfk; 1 fhjk; (1200 nr.kP.)4 fhjk; 1 Nahrid

nghd; epWj;jy; tha;g;ghL

4 ney; vil 1 Fd;wpkzp2 Fd;wpkzp 1 kQ;rhb1 kQ;rhb 1 gz tpil5 gz tpil 1 foQ;R8 gz tpil 1 tuhf vil4foQ;R 1 f/R4f/R 1 gyk; gz;lq;fs; epWj;jy; tha;g;ghL

32 Fd;wpkzp 1tuhfndil10 tuhfndil 1 gyk;40 gyk; 1 tPir6tPir 1 J}yhk;8 tPir 1 kzq;F20 kzq;F 1 ghuk;

gz;lq;fs; epWj;jy; tha;g;ghL32 Fd;wpkzp 1 tuhfndil10 tuhfn 1 gyk;40 gy 1 tPir6tPir 1 J}yhk;8 J}yhk; 1 kzq;F20 kzq;F 1 ghuk;

epWj;jysit1kQ;rhb 260 kp.fp.1 foQ;R 5.1 fp.1 mTd;]; 30 fp.1 J}f;F 1.7 fp.

njwpg;gsT1 ehs; 60 ehopif/24 kzp1 kzp 2.5 ehopif 60 epkplq;fs;1 ehopif 24 epkplq;fs;1 ehopif 60 tpehopif1 epkplk; 2.5 tpehopif1 tpehopif 60 tpypg;jk;

nga;jy; tha;g;ghL

360(300) ney; 1 nrtpL5 nrtpL 1Mohf;F2 Mohf;F 1 cof;F2 cof;F 1 chp3 cof;F 1%Tof;F 2 chp 1ehop(gb)8 ehop(gb) 1FWzp (kuf;fhy;)5 kuf;fhy; 1 giw80 giw (m)400 kuf;fhy; 1 fhpir

gilfspd; msT

1 ge;jp 1 ,ujk; (Njh;);;;; 1 Mid 3 Fjpiu 5 fhyhs;

3 ge;jp 1 NrdhKfk;3 NrdhKfk; 1 Fy;kk;3 Fy;kk; 1 fzfk;3 fzfk; 1 thfpdp3 thfpdp 1 rkhf;fpak; 10 rkhf;fpak;1 mf;FNuhzp8 mf;Fnuhzp1 Vfk;8 Vfk; 1 Nfhb8 Nfhb 1 rq;fk;8 rq;fk; 1 tpe;jk;8 tpe;jk; 1 FKjk;8 FKjk; 1 gJkk;8 gJkk; 1 ehL8 ehL 1 rKj;jpuk; 8 rKj;jpuk; 1 nts;sk;

fhy tha;g;ghL2 fz;zpik 1 nehb2 if nehb 1 khj;jpiu2 khj;jpiu 1 FU2 FU 1 caph;2 caph; 1 rzpfk;12 rzpfk; 1 tpehb60 tpehb 1 ehopif2 1/2 ehopif 1 Xiu3 3/4 ehopif 1 K$h;j;jk;2 K$h;j;jk; 1 rhkk;

jkpo; khjq;fs;

rpj;jpiu Nklk; Nkok; 30 ehs;

itfhrp ,ltk; tpil 31 ehs;

Mdp kpJdk; Mlit 31 ehs;

Mb fw;flfk; flfk; 31 ehs;

Mtzp rpq;fk; klq;fy; 31 ehs;

Gul;lhrp fd;dp fd;dp 30 ehs;

Ig;grp Jyhk; Jiy 29 ehs;

fhh;j;jpif tpUr;rpfk; esp 29 ehs;

khh;fop jDR rpiy 29 ehs;

ij kfuk; Rwtk; 29 ehs;

khrp Fk;gk; Fk;gk; 29 ehs;

gq;Fdp kPdk; kPdk; 30 ehs;

Afq;fs;1 fw;gk; 1000 rJh; Afk;1 kDte;juk; 71 rJh; Afk;1000 rJh; Afk; 4 Afq;fs;4 Afq;fs; 43> 20> 000 Mz;Lfs;fpUj Afk; 4 43 >20> 000 Mz;Lfs;jpNuj Afk; 3 43> 20 >000 Mz;Lfs;Jthgu Afk; 2 43> 20 >000 Mz;Lfs;fypAfk; 1 43> 20>000 Mz;Lfs;1 Mz;L tl;lk; 60 Mz;Lfs; (rpyh; ,ij 64 Mz;Lfs; vdf;nfhs;Sk; tof;fKk; ,Uf;fpwJ)1 Mz;L 12 khjq;fs;

Facts about 100 (Hundred) A person who lives to be 100 years is called a centenarian. C is the Roman numeral for 100 C comes from the Latin word Centrum. 10 ten dollar buills are 100 dollars. The 100 years War started between and france in 1336. A hundred Watt light bulb lasts for 750 hours. The place value system was developed in India in 100 B.C. Cats makes 100 different sounds. In Canada 100 different language are spoken. 100 mile per hour winds is a category 2 huricane. One tablespoon of peanut butter has 100 calories. The average hippo weighs 100 pounds at birth. The Wright brothers invented the first aeroplane over 100 years ago. 100 written in the binary system is 1100100. The standard prefix for 100 is hecto. The sum of the first 9 prime numbers is 100.

The only way to learn mathematics is to do mathematics. Paul HalmosDo you know? Taj Mahal in Agra is a symmetrical monument. Symmetry reers to the exact match in shape and size between two halves of an object . If we fold a picture in half and both the halves lef half and right half match exactly then we say that the picture is symmetrical.Golden ratio: Golden Ratio is a special number approximately equal to 1.6180339887498948482 . We use the Greek letter Phi () to refer to this ratio. Like Phi the digits of the Golden Ratio go on forever without repeating. Golden Rectangle: A Golden Rectangle is a rectangle in which te ratio of the length to the width is the Golden Ratio. If which of the Golden Rectangle is 2 ft. long, the other side is approximately = 2(1.62) = 3.24 ft.Golden segment: It is a line divided into 2 parts. The ratio of the length of the 2 parts of this segment is the Golden Ratio. = Triangular numbers

1 3 6 10 15 21 28 36

The set of numbers 1, 3, 6, 10, generated by triangular arrays of dots.

The purpose of computing is insight, not numbers. Hamming.