Singapore Mathematical Olympiad Training Handbook - Sec 1

Transcription

1 Develop The Mths Genius in You Singpore Mthemticl Olympid Trining Hndbook - Sec Includes Questions from Includes Questions from other Olympids

2 Der young students of mthemtics Mthemtics is wonderful subject It is one of the most useful wys to develop your mind The mteril in front of you hs been developed over the yers in trining tlented pupils in this subject in top secondry girl school If you re Primry 6 or even Primry 5 pupil who is seeking chllenges or Secondry pupil who is looking for wys to develop your mthemtics tlent, look no further Pick up pencil nd hve go t it This hndbook contins copyrighted mteril so it should strictly be for your personl use By the end, I hope you enjoy wht I hd put together here for you Cheers Mr Ang K L, 0

3 Contents Worksheet From Arithmetic to Algebr Formulting equtions 5 3 Algebr with Arithmetic 4 Systems of Eqution 6 5 Rtionl Numbers nd some of its opertions 6 Appliction of bsic lgebr 8 7 Odd nd even integers 35 8 Prime nd Composite Numbers 38 9 Divisibility 4 0 Rtio nd its properties 47 Perfect Squres 53 Modulr Arithmetic 56 3 Lines nd ngles 60 4 Prllel Lines 63 5 Tringles 68 6 Are 73 7 Solving Diophntine Equtions 79 Tropicl Revision Rtionl Numbers 83 Integrl Epressions 85 3 Solve liner equtions 87 4 Solving dul vribles liner Equtions 89 5 Liner inequlities 90 6 Prllel lines 9 7 Tringles 94 8 Problem solving 97 Problem Solving Prctice # 00 Prctice # 04 3 Prctice # Prctice # Prctice #5 0 6 Prctice #6

4 7 Prctice #7 4 8 Prctice #8 6 9 Prctice #9 9 0 Prctice #0 Prctice # 3 Prctice # 5 3 Prctice #3 8

5 -: :- From Arithmetic to Algebr Wht is Algebr? Algebr is study of number properties in the form of lphbeticl representtion E C? The verge of three numbers A, B, nd C is If the verge of A nd B is b, wht is the vlue of E The cost of shirt, ht nd pir of shoe is $ A shirt is $b more epensive thn pir of shoe A shirt is $c more epensive thn ht Find the cost of pir of shoe The opertions in lgebr re the sme s those in rithmetic

6 -: :- E 3 There re two piles of printing ppers on tble The first pile is kg more thn the second pile If b kg re used in ech pile, the first pile will be times of second pile Find the weights in ech pile E 4 To complete job, A lone took dys, B lone took b dys Find the number of dys it would tke both of them to complete the job together

7 -: 3 :- E 5 To complete job, A lone took dys, B lone took b dys Now tht A lone hd completed c dys c, then B lone complete the rest of the job How mny dys would B tke to complete the rest of the job? E 6 There re number of chicken nd rbbits, with b number of legs How mny rbbits re there?

8 -: 4 :- Prctices There re two bskets of pples If pples were moved from first bsket to the second bsket, then the two bskets would hve the sme number of pples If b pples were moved from second bsket to the first bsket, then the first bsket would hve twice s mny s the second bsket Find the number of pples in ech bsket st 4 3b nd 3b After Mth test, the top 0 pupils hd n verge of The top 8 pupils hd n verge of b The ninth pupil hd c more thn the tenth pupil Find the score of the tenth pupil 0 8b c 3 There re some 4 nd 8 stmps, with totl mount of $ If there re b pieces more 8 stmps thn 4 stmps, find the number of 8 stmps nd the number of 4 stmps re there 5 b 3 4 A fst cr nd slow truck deprted from town X nd town Y respectively, towrds ech other It took hours for the fst cr to rech town Y The slow truck took b hours to rech town X If the fst cr trvelled m km more thn the slow truck in n hour, how long would it tke for the two to meet on their journey? b b 5 To complete job, Mr A lone nd Mr B lone took nd b dys respectively To complete this job, Mr A did few dys, then Mr B took over to complete the rest It took c dys b c in totl to get it done Find the number of dys ech took to complete this job c b b

9 -: 5 :- Formulting Equtions To solve problems with lgebr will generlly require the forming of eqution(s) An eqution is n epression of two equl quntities tht re divided by the sign = In order for this to be possible, we will lern how to trnslte from words into lgebr E A horse nd donkey met on their wy The donkey sid to the horse: If you trnsfer one bg to me, my lod would hve been twice of your lod The horse replied: If you trnsfer one bg to me, our lod would hve been even Find the number of bgs on the donkey Here re the steps to tke to form n eqution: Let the number of bgs on the donkey be The donkey sid to the horse: If you trnsfer one bg to me, my lod would hve been twice of, donkey, horse your lod (we cn then tell tht the horse ctully hd) + The horse replied: If you trnsfer one bg to me, donkey our lod would hve been even, (From the lst sttement, we know tht the number of bgs on the horse is the sme s the number of bg tht is on the donkey)

10 -: 6 :- (This is your first eqution) (This is the concept of blncing) (Agin, pply the blnce concept) 7 7 (Once gin, blncing) 7 Cn you formulte this problem differently? Let s try!

11 -: 7 :- E A, B, C, nd D together, hve 45 books If A hs less, B hs more, C hs double, nd D is hlved, then ech would hve the sme number of books How mny books hs A? Let ech would hve the sme number of books Be A hs less, then A ctully hs B hs more, then B ctully hs C hs double, then C ctully hs D is hlved, then D ctully hs A, B, C, nd D together, hve 45 books Use this to form n eqution: 45 We cn then simplify this eqution into: (Remember the concept of blncing) 0

12 -: 8 :- E 3 A group of students ws to clen up to two res in their school Are A ws times of Are B In the morning (hlf of dy), the number of students clening Are A ws 3 times tht of the number of students in Are B In the fternoon (nother hlf of dy), 7 of the students worked in Are A while the rest of them in Are B At the end of the dy, Are A ws done, but Are B still needed 4 students to work one more dy before it ws done How mny were there in this group of students? E 4 Jug A contined litres of pure honey, nd Jug B contined 5 litres of pure wter Some honey from Jug A ws poured into Jug B, the miture ws well stirred Net, some miture from Jug B ws poured into Jug A At the end, Jug A still contined 65% of honey by volume nd Jug B contined 5% of honey by volume If the totl volume remined the sme, how much hd the miture been poured into Jug A?

13 -: 9 :- Prctices There re two wrehouses; with the first one hs three times the number of TV sets thn the second one If 30 sets were trnsferred from the first to the second, then the second one would hve 4 tht of the 9 first one Find the number of TV sets in the second one 30 There were 40 blck chocolte brs nd white chocolte brs on shelve After one qurter of the blck chocolte brs ws sold, the storekeeper dded nother 50 white chocolte brs on the shelve Then, the number of white chocolte brs would be twice the number of blck ones Find the number of blck chocolte brs t first 76 3 Mr A nd Mr B were to deprt from the sme plce to town X Mr A wlked t speed of 5 km/h After he hd deprted for one nd hlf hour, Mr B cycled to town X It took Mr B 50 minutes to rrive t town X together with Mr A Find the speed of Mr B 4 4 A nd B deprted together to town Y, B on foot, nd A by bicycle A s speed is km/h more thn thrice of B Upon rriving t town Y, A rested for n hour before returning On the return trip, A met B 3 when B hd lredy wlked for two nd hlf hours If town Y ws 4 km wy from their deprture 4 point, find the speeds of the two nd how fr hd they ech trvelled before they met gin? 3,4 5 A motorist deprted t 9 m from town A to town B He plnned to rrive t noon An hour lter, he relized tht he would be lte by 0 minutes with his current speed As such, he incresed his speed by 35 km/h nd in so doing, rrived t ectly noon Find the originl speed of the motorist nd 0,700 the distnce between the two towns 6 The numbers of pupils in two groups re in the rtio of 4 : If 5 pupils re trnsferred from the first group to the second group, then, there will hve sme number of pupils in ech group How mny pupils re to be trnsferred from the first group to the second group so tht the rtio becomes 3 : 7? 5

14 -: 0 :- 7 There re two cndles, one thick nd the other one thin, but re of equl length The thick one cn lst 5 hours The thin one cn lst 4 hours If the two cndles re lighted together, how long will it tke for the thick one to be 4 times tht of the thin one? 3 3 4

15 -: :- Algebr with Arithmetic In solving mny mthemticl problems, the rithmetic pproch seek to develop better understnd of the problem over the lgebric counterprt In combining the use of the two pproches, one cn usully find solution to problem much esier E A cr is trveling from town X to town Y If the speed of the cr is incresed by 0%, it rrives t town Y one hour erlier thn s plnned If it hs, t first, trvelled for 0 km with the originl speed, then increses its speed by 5% for the rest of the journey, it will rrive 40 minutes erlier insted Find the distnce between town X nd town Y Method, Arithmetic pproch Method, Algebric pproch,

16 -: :- E A job cn be done by Mr A lone in 9 dys, Mr B lone in 6 dys Now tht Mr A hs done 3 dys of the job, how mny dys will it tke Mr B to complete the job, without Mr A? Method, Arithmetic pproch Method, Algebric pproch, E 3 For project, tem A cn complete the project in 0 dys Tem B cn complete the project in 30 dys Now tht both tem re working on the project But tem A hs two rest dys, nd tem B hs 8 dys of rest Find the number of dys it will tke them to complete the project

17 -: 3 :- E 4 A project will tke 63 dys by tem A, nd then nother 8 dys by tem B to complete If both tems re to work on this project together, it will tke 48 dys to complete If tem A is to work 4 dys, how mny dys will it tke tem B lone to complete the rest of this project? E 5 There re two wter filling pipes, A nd B nd one drin pipe, C connected to pool It tkes 3 hours to empty full pool with ll 3 pipes open It tkes just one hour to empty this pool with pipe A nd pipe C only It tkes 45 minutes to empty this pool with pipe B nd pipe C only If the filling rte of pipe A is 3 m more thn pipe B, find the fill rte nd drin rte of ech pipe min

18 -: 4 :- E 6 A jug contined some liquid(wter nd lcohol miture) After cup of wter ws dded, the concentrtion of lcohol in the jug becme 5% After nother cup of pure lcohol ws dded into the jug, the concentrtion of lcohol ws 40% How mny cups of liquid were there in the jug t first? mount of lcohol (Concentrtion of lcohol by volume ) mount of liquid E 7 Two tems were working on writing book Tem A wrote 3 of the book in 4 dys Then tem B joined the project With tem A, they finlly completed the book in 3 dys If tem B wrote 75 pges of the book, find the number of pges in this book

19 -: 5 :- Prctices A project will tke Mr A nd Mr B dys to complete Now tht both of them work for 4 dys, with the rest to be completed by Mr A in 0 dys, find the number of dys ech tke to complete this project by himself 5,60 In nother project, if Mr A works on it for dys nd Mr B works on it for 5 dys, 5 4 of the project will be completed But if Mr A works on it for 5 dys nd Mr B works on it for dys, then 9 of the 60 project will hve been completed Find the number of dys ech tke to complete this project by him lone 0,30 3 A tnk is filled from empty to full by pipe A in minutes It only tkes pipe B 5 minutes to drin it completely Pipe C tkes 6 minutes to fill this tnk If pipe A is open to fill n empty tnk for the first few minutes before pipe B nd C re open, it will tke 8 minutes to fill this tnk How long hs pipe A been open before the other two pipes re open? 3 4 The mount of work done by Mr B in dy took Mr A one-third of dy to do The mount of work done by Mr C in dy took Mr B 3 of dy to do Now, ech dy, of them were to work on 4 project It took Mr A 4 dys, Mr B 3 dys nd Mr C 3 dys to complete this project Find the number of 3 dys Mr A lone took to complete this project 5 4

20 -: 6 :- System of Equtions When there re or more unknowns, it often requires or more equtions to be set up The wys of solving these equtions re the lesson for tody E A frction, fter being simplified, is 3 If integer is dded to both the numertor nd its 8 denomintor of this frction, it becomes If one is dded to this integer, nd the new integer is subtrcted from both the numertor nd the denomintor of this frction, it becomes 9 5 Find this frction Method, Algebric pproch, Method, Arithmetic pproch,

21 -: 7 :- E Mr A, Mr B, nd Mr C took prt in bicycle rce Mr A finished minutes erlier thn Mr B Mr B finished 3 minutes before Mr C If Mr A ws 5 km/h fster thn Mr B, nd Mr B ws km/h fster thn Mr C, find the distnce of their rce Method, Algebric pproch, Method, Arithmetic pproch,

22 -: 8 :- E 3 A red bllpoint pen costs 9 cents, nd blue bllpoint pen costs cents Now tht I py totl of $80 for 6 pens, how mny re blue pens? E 4 Clerk A tkes 6 hours to type report nd clerk B tkes 0 hours to type the sme report If clerk A strts to type for few hours then hnd over the rest of the typing to clerk B, it will tke 7 hours in totl to complete Find the number of hours ech tkes to type this report?

23 -: 9 :- E 5 Two iron ores, the first ores contin 68% of iron nd the second ores contins 63% of iron Now tht 00 tones of 65% of iron ores re required, how much of ech ores is to be used?

24 -: 0 :- Prctices A delivery order took fleet of trucks over number of dys to fill If 6 fewer trucks were used, then it would etend 3 dys to complete the order If 4 dditionl trucks were used, then it would hve shortened one dy to complete Find the number of dys nd the number of the truck in this fleet 5,6 There were 360 door gifts for Founder s dy to be ssembled over the weekend Tem A produced % of its quot; tem B produced 0% of its quot As result, there were 400 door gifts ssembled 4,6 in totl Find the number of gift in ecess of the quot from ech tem 3 30 English books nd 4 Mth books cost $8340 in totl An English book costs 44 cents more 74 thn Mth book Find the cost of n English book 4 A chemicl of 80% concentrtion is to be mied with the sme chemicl of 90% concentrtion to produce 84% concentrtion chemicl For 500 litres of this mied chemicl, find the mount used by 300,00 ech chemicl 5 Alcohol of 7% concentrtion is mied with lcohol of 58% concentrtion to mke 6% miture If 5 litres more of ech lcohol re dded to this miture, the concentrtion becomes 635% Find the,30 mount of ech lcohol used in the first miture

25 Numbers nd number line -: :- Rtionl Number nd some of its opertions E Observe the digrm, A B - b C c, b, nd c re vlues correspond to point A, B, nd C respectively Arrnge scending order b, b nd c in E By dding opertors + or - between the numbers , Wht is the lest nonnegtive vlue E 3 Find the sum of

26 -: :- E 4 Evlute E 5 Compre the vlue of nd

27 -: 3 :- Prctices Refer to the digrm below, two points A nd B re on number line correspond to vlue nd b 5b If, which vlue, or is lrger? 5b A B b Clculte Derive the formul for n n n 4 In this number pttern: 3 ;,, ;,,,, ;, () which position is 7 0? 33 (b) which number is on the 400 th position? Clculte Given tht O is the origin on number line Point A nd B re positions on nd Let P be the mid-point on AB, P be the mid-point on AP,, P 00 be the mid-point on AP 99, find the vlue of P P P3 P00 7 Evlute ( ) () () 3 ( ) 99 () 00 8 Find the sum of Evlute

28 -: 4 :- m 0 The sum of is in its lowest n terms Find the vlue of m + n By number line Compring Rtionl numbers Number, b, c re on number line s shown in the digrm below - 0 b c Of these 4 numbers,,, c b, c, which is the lrgest? On the number line, number on the left is lwys less thn number on its right All positive number is lwys lrger thn 0, zero All negtive number is less thn 0 Therefore ll positive number is lrger thn ll negtive number In the digrm bove, 0, 0 b c c, cb0 0, 0, Of these numbers,, c b, c, is the lrgest Alterntive Method, Let, b, 4 3 c, 4,, 3 c b, c 4 4

29 -: 5 :- Therefore is the lrgest Prctice, b Q, s shown on number line below A B b 0 C b b D b b, b, c re vlue on the number line below, find the lrgest of these 3 numbers b, c b, c b c 0 3, b re shown on number line below, which one of the inequlity is incorrect? b A b B b C b D b

30 -: 6 :- By the difference of the numbers E If, b re rtionl, nd b 0, then wht is the reltive sizes of, b, b? Given two numbers nd y, if y 0, then y; if y 0, then y; if y 0, then y b b b 0, therefore b b b b 0, therefore b b b Prctice Compring nd 3 Compring A nd B Compring nd By division Compring 9 99 P nd Q 90 9 b nd b 0 b b nd b 0 b b b

31 -: 7 :- P Q P Q Prctice Compring 555 3, 444 4, nd Given tht m 0, n 0, compring m, mn, mn 3 If n, compring n n, n n, n n 4 If b < 0, then then the reltion in sizes of ( b) nd ( + b) is () ( b) < ( + b) (b) ( b) = ( + b) (c) ( b) > ( + b) (d) Not determined

32 -: 8 :- Appliction of bsic Algebr Find the sum of We cn re-rrnge the bove epression into There is pttern of n n 4n 4n 4n 4n Therefore, Prctices Find the sum of Find the product of Find the sum of

33 -: 9 :- Evlute simple lgebric epression(i) Clculte Let, nd 3 4 ; 3 4 ; then Try it out! From 009, subtrct hlf of it t first, then subtrct 3 of the remining number, net subtrct 4 of the remining number, nd so on, until 009 remining number? of the remining number is subtrcted Wht is the Evlute Prctices Given tht 4 y b c, when 5, y 3 Find the vlue of y when 5

34 -: 30 :- Given tht when 7, the vlue of the epression 5 b 3, when 7 5 b 8 is 4 Find the vlue of 5 3 Given the epression b c hs vlue of 8 nd when 3 or 0 respectively, find the vlue of the epression when 3

35 -: 3 :- Evlute simple lgebric epression(ii) b Given tht 7, find the vlue of b b b b 3 b b Since 7, then we hve b b b 7 b b b b b 3 b b 3 b Prctice If the epression y 3y 7, then find the vlue of 4y 6y 9 Given tht , b , nd c000 00, then find the vlue of b b c c 3 Let 3 5 b 6, find the vlue of bb b b

36 If b, find the vlue of -: 3 :- Evlute simple lgebric epression(iii) b b b, b b b b b b b b b b b b Prctice If y z, nd y, then find the vlue of y z y z If y, nd y y 4 0, find the vlue of y y 3 Given tht y, find the vlue of 3 y 6 4 y 8 4 If 4, find the vlue of z 4z z z z

37 -: 33 :- Evlute simple lgebric epression(iv) If y z, y, then find the vlue of y z y z Given tht y z, y, we cn choose, y 3, nd z to find the vlue of the epression y 3 3 z y z 3 Prctice Given tht y, find the vlue of 3 y 6 4 y 8 Given tht f, y 3 y m, nd f, 8, find the vlue of 3, f b c c b 3 If b c 0, nd 0, find the vlue of b c bc b c c c b b b c c b

38 -: 34 :- Evlute simple lgebric epression(v) y z If, nd 3 y z 8, find the vlue of 5y 3z y z y z From, let k Then we hve 3k, y 4k, z 5k 3 y z 8 k k k k 3, 9, y, z 5 y z Prctice y z If, nd 4 5y 0, find the vlue of 5y z If y z, nd y z, find the vlue of 3y 4z 3 3 If y z b b c c, find the vlue of y z 4 SMO 009 Junior Pper Given tht b nd b 0, find the vlue of b + b b

39 -: 35 :- Odd nd Even Integers Integers cn be divided into two sets of number The set with numbers tht cn be divided by is clled Even; nd the rest tht cnnot be divided by ectly, is clled Odd An Even integer cn be denoted s n, where n is n integer An Odd integer cn be denoted s n, where n is n integer An integer must be either Odd or Even, but not both Some of the other properties: () Odd Even () Odd + Even = Even + Odd = Odd; Odd Even = Even Odd = Odd (3) Even + Even = Even; Even Even = Even (4) If beven, then t lest one of the fctors is even n n even (5) The product of two consecutive integers must be even, (6) If the sum or difference of integers is odd, then there must be t lest n Odd integer in the sum or difference (7) If the sum or difference of integers is even, then the number of odd integers must be even (8) If the sum or difference of integers is odd, then the number of odd integers must be odd (9) If the product of integers is odd, then ll the numbers in the product must be odd (0) If the product of integers id even, then there must be t lest one even integer b b c c E If bcre,, rndom integers, then mong the three numbers,,, there re (A) ll non-integers (C) t lest two integers (B) t lest n integer (D) ll integers E Given tht the sum of 00 positive integers is 0000, the number of odd integers is more thn the number of even integers, wht is the most number of even integers? (A) 49 (B) 48 (C) 47 (D) 46

40 -: 36 :- E 3 Given tht nd b re consecutive integers nd tht c b, prity of N? N b c, wht is the (A) Odd (B) Even (C) mybe Odd or Even (D) none of the bove E 4 There re n number:,, 3,, n Ech of these numbers is either or - If 0, wht must be truth bout n? 3 n n n (A) Even (B) Odd (C) multiple of 4 (D) cnnot tell t ll E 5 Given tht positive integers p, q, p q re primes, nd lso tht p q is even, evlute the vlue of p 3 q E 6 In this number pttern:,,5,3,34,89,, strting from the second number, the sum of ny two djcent number is equl to three times the middle number Wht should be the prity of the 003 rd number?

41 -: 37 :- E 7 Given three integers,, must be even y z with two odds nd one even, prove tht y z 3 Prctices Given 003 consecutive positive integers:,,3,4,,003, if either + or opertor is dded in between ny two numbers, will the result be odd or even? even In Mth competition, there re 40 questions A correct nswer scores 5 points; nil return scores point; nd wrong nswer deducts point Should the totl score of ll the competitors be even or odd even 3 30 books re to be pcked into 5 boes Ech bo must hve odd number of books How cn this be done? impossible 4 Is it possible to rrnge these 0 numbers:,,,, 3, 3, 4, 4, 5, 5, on line such tht there is number in between the two s; there re two numbers in between the two s; ; there re five numbers in between the two 5 s? impossible 5 Is it possible to visit ll the 6 rooms just once without re-entering ny one of them? impossible

42 -: 38 :- Prime nd Composite Numbers Among ll the positive integers, is the only one tht hs only one positive fctor, nd tht is itself All positive integers greter thn hve t lest two positive fctors If n positive integer hs only two fctors, tht is, nd the number itself, then this integer is clled Prime number, or Prime in short All non-prime integers re collectively clled Composite Number, or Composite in short By definition, is NOT prime There is only one even prime,, which is lso the smllest prime The smllest composite is 4 is lso NOT composite All positive integer, other thn, cn be prime fctorise N p p p3 p k 3 k 3 where 3 k p, p, p,, p re distinct primes k The number of positive fctor of N cn be found with this epression: Number of positive fctor of N = 3 k E If p nd 3 p 5 re prime, wht is 5 p 7? (A) Prime (B) Prime or Composite (C) Composite (D) Not prime or composite E Given three primes pqrtht,, stisfy p q r nd p q, find the vlue of p (A) (B) 3 (C) 7 (D) 3

43 -: 39 :- E 3 Given tht n is positive integer such tht n 3 nd n 7 re both primes Find the reminder when n is divided by 3 E 4 If positive integers n n, nd n n n n 9, find the vlues of n, nd of n E 5 If pqr,, re primes, nd 66 p q r 986, find the vlue of p q r E 6 If p is prime not less thn 5, nd p is lso prime, prove tht 4p is composite

44 -: 40 :- Prctices Given y, re primes, find the number of ordered pirs of eqution y 999 (A) pir (B) pirs (C) 3 pirs (D) 4 pirs Given tht mn, re distinct primes, nd p m n mn with p being the minimum vlue, evlute the epression m n p 3 Given tht pq, re primes nd p m n, q mn, where mn, re positive integers, evlute the epression p m p n q n q m 4 If mn, re primes tht stisfy 5m7n 9, find the vlues of m n 5 Given tht p, p, p 6, p 8, p 4 re primes, find the number of p 6 SMO question 009 Let p nd q represent two consecutive prime numbers For some fied integer n, the set {n, 3n 9, 38 5n, 7n 45} represents {p, p, q, q}, but not necessrily in tht order Find the vlue of n

45 -: 4 :- Divisibility A divisibility rule is method tht cn be used to determine whether number is evenly divisible by other numbers Divisibility rules re shortcut for testing number's fctors without resorting to division clcultions Although divisibility rules cn be creted for ny bse, only rules for deciml re given here The rules given below trnsform given number into generlly smller number while preserving divisibility by the divisor of interest Therefore, unless otherwise noted, the resulting number should be evluted for divisibility by the sme divisor For divisors with multiple rules, the rules re generlly ordered first for those pproprite for numbers with mny digits, then those useful for numbers with fewer digits If the result is not obvious fter pplying it once, the rule should be pplied gin to the result Divisor Divisibility Condition Emples Automtic Any integer is divisible by The lst digit is even (0,, 4, 6, or 8),94: 4 is even 3 The sum of the digits is divisible by 3 For lrge numbers, digits my be summed itertively 405: = 9, which clerly is divisible by 3 6,499,05,854,376 sums to 69, = 5, + 5 = 6, which is clerly divisible by 3 The number obtined from these emples must be divisible by 4, s follows: 4 If the tens digit is even, the lst digit is divisible by 4 (0, 4, 8) If the tens digit is odd, the lst digit plus is divisible by 4 (, 6) If the number formed by the lst two digits is divisible by 4 68: 6 is even, nd 8 is divisible by 4 5,496: 9 is odd, nd 6+ is divisible by 4,09: 9 is divisible by 4 5 The lst digit is 0 or 5 490: the lst digit is 0 It is divisible by nd by 3 4: it is divisible by nd by 3 6 Add the lst digit to four times the sum of ll other digits 98: ( + 9) = 48 The number obtined from these emples must be divisible by 7, s follows: 7 Form the lternting sum of blocks of three from right to left,369,85: = 483 = 7 69

46 -: 4 :- Double the number with the lst two digits removed nd dd the lst two digits 364: (3 ) + 64 = 70 Add 5 times the lst digit to the rest 364: 36 + (5 4) = 56 Subtrct twice the lst digit from the rest 364: 36 ( 4) = 8 The number obtined from these emples must be divisible by 8, s follows: 8 If the hundreds digit is even, emine the number formed by the lst two digits If the hundreds digit is odd, emine the number obtined by the lst two digits plus 4 64: 4 35: = 56 Add the lst digit to twice the rest 56: (5 ) + 6 = 6 9 The sum of the digits is divisible by 9 For lrger numbers, digits my be summed itertively Result t the finl itertion will be 9,880: = 8: + 8 = 9 0 The lst digit is 0 30: the lst digit is 0 The number obtined from these emples must be divisible by, s follows: Form the lternting sum of the digits 98,08: = Add the digits in blocks of two from right to left 67: = 33 Subtrct the lst digit from the rest 67: 6-7 = 55 It is divisible by 3 nd by 4 34: it is divisible by 3 nd by 4 Subtrct the lst digit from twice the rest 34: (3 ) 4 = 60 The number obtined from these emples must be divisible by 3, s follows: 3 Add the digits in lternte blocks of three from right to left, then subtrct the two sums,9,7: ( + 7) + 9 = 637 Add 4 times the lst digit to the rest 637: 63 + (7 4) = 9, 9 + ( 4) = 3 It is divisible by nd by 7 4: it is divisible by nd by 7 4 Add the lst two digits to twice the rest The nswer must be divisible by 7 364: (3 ) + 64 = 70

47 -: 43 :- 5 It is divisible by 3 nd by 5 390: it is divisible by 3 nd by 5 The number obtined from these emples must be divisible by 6, s follows: 6 If the thousnds digit is even, emine the number formed by the lst three digits If the thousnds digit is odd, emine the number formed by the lst three digits plus 8 Sum the number with the lst two digits removed, times 4, plus the lst two digits 54,76: 76 3,408: = 46 76: ( 4) + 76 = 80 Subtrct the lst two digits from twice the rest 33: ( 3) - 3 = 5 7 Alterntely dd nd subtrct blocks of two digits from the end, doubling the lst block nd hlving the result of the opertion, rounding ny deciml end result s necessry 0,98,65: (65 - (98 )) : + 40 = - 55 = 55 = 5 7 Subtrct 5 times the lst digit from the rest : - ( 5) = 7 8 It is divisible by nd by 9 34: it is divisible by nd by 9 9 Add twice the lst digit to the rest 437: 43 + (7 ) = 57 0 It is divisible by 0, nd the tens digit is even If the number formed by the lst two digits is divisible by 0 360: is divisible by 0, nd 6 is even 480: 80 is divisible by 0 E The gretest integer tht cn divide the sum of ny three consecutive integers is: (A) (B) (C) 3 (D) 6 E A si-digit integer 568bc is divisible by 3, 4, 5 Find the smllest number bc

48 -: 44 :- E 3 Prove tht is multiple of 0 E 4 Find the reminder when is divided by 9 E 5 Given tht A , find the lst three digits of A E 6 Given nturl numbers from to 995 inclusively, find the number of from these 995 numbers tht 995 cn divide ectly995

49 -: 45 :- Prctices Find the number of 7-digit number 87y 6 tht is multiple of 7 If n is positive integer, n3, n 7 re primes, find the reminder when n is divided by 3 3 Find the lst digit of Let n be the lst two digits of 7 n, find the vlue of Prove tht cn be divided ectly by 5 6 SMO Question 995 A nturl number greter thn gives the sme reminder (not zero) when divided b 3, 5, 7 nd Find the smllest possible vlue of this nturl number 7 SMO Question 998 Find the smllest positive integer n sch tht 00 n 00 nd n n n n re divisible by 0 8 SMO Question 999 When the three numbers 38, 596 nd 49 re divided by positive integer m, the reminders re ll equl to positive integer n Find m + n 9 SMO Question Find the smllest positive integer k sch tht k is divisible by 7

50 -: 46 :- 0 SMO Question 003 n n n n How mny integers n between nd 003 (inclusive) re there such tht is divisible by 5? n SMO Question 009 m nd n re two positive integers stisfying m n 40 Find the number of pirs of (m, n) such tht their product is divisible by 33

51 -: 47 :- Rtio nd its properties If John hs 75 cents nd Mry hs 50 cents, the rtio of the mount of money of John to Mry is 75 : 50 or 3 : You know this very well If we let the mount of money tht John hs be J nd the mount of money tht Mry hs be M, then here is how we cn epress this rtio J: M 3: Alterntively, we cn lso write the rtio s J 3 M Now consider this: let sy tht the rtio of the mount of money of John nd Mry be 3 :, wht is the mount of money of ech of them? J 3 The rtio is The mount of money of John is 3 M J m ; the mount of Mry is M m, where m is the common unit We cn see tht J M m Therefore this rtio of John nd Mry cn be epressed s 3 J M 3 These re ll equivlent sttement of the sme thing J: M 3: J 3 M J M 3 If the rtio of A: B : C : b: c, it will be esier to epress it s A B C, rther thn b c A nd B b B C b c Cn you prove this reltionship, A B C? b c

52 -: 48 :- In generl, given tht : b c : d d bc (i) (ii) c b b d d c (iii) Given c b d, we hve b c d Cn you prove this? b d (iv) Given c b d, we hve b c d Cn you prove this? b d (v) Given c, we hve b c d Cn you prove this? b d b c d (vi) Given c e n c e n ; if b d f m 0 b d f m b d f m The bove etension of the bsic rtio cn be helpful in solving rtio relted problems E Given tht b 3, which of the following is/re not true sttement? (B) k, b 3k k 0 (A), b 3 (C) 3 b (D) b 3 E Fill in the blnk (A) Given tht 3 b 5, b b (B) Given tht 3 4 b 7, 3b 4 b c e 3 (C) Given tht b d f 4, 3 c e 3b d f

53 -: 49 :- y z y yz z (D) Given tht, 3 4 y z E 3 Which of the following is/re true sttements, (A) if c, then c m ; (B) if c, then b d b d m b d b c ; d (C) if c b d, then c b c ; (D) if b d b c, then c d b d E 4 (A) Given y, find the vlue of 3y y y (B) if, where, 0 6 5y y 0 y, find the vlue of 3 y 3y

54 -: 50 :- y z E 5 Given tht : y 3:5 nd y: z :3, find the vlue of y z E 6 Given tht 3y z 0, 3 y 6z 0 nd yz 0, find the vlue of, (A) : y : z, (B) y z y z

55 -: 5 :- Prctices Given tht c b, find the vlue of b b c c Given tht 3, find the vlue of 3 y y y y y 3 Given tht b b, find the vlue of b b 4 Given tht b b 3 bc, 3 b c, c 003 c, find the vlue of 004 bc b bc c 5 Given tht c, prove tht b d b c d b cd 6 Given tht y z, show tht b c y z y z b c bc 3 7 SMO Question 998 Three boys, Tom, John nd Ken, greed to shre some mrbles in the rtio of 9: 8 : 7 respectively John then suggested tht the should shre the mrbles in the rtio 8: 7: 6 insted Who should then get more mrbles thn before nd who world get less thn before if the rtio is chnged?

56 -: 5 :- 8 SMO Question 009 There re two models of LCD television on sle One is 0 inch stndrd model while the other is 0 inch widescreen model The rtio of the length to the height of the stndrd model is 4: 3, while tht of the widescreen model is 6: 9 Television screens re mesured b the length of their digonls, so both models hve the sme digonl length of 0 inches If the rtio of the re of the stndrd model to tht of the widescreen model is A: 300, find the vlue of A

57 Perfect squres re nturl numbers:, numbers -: 53 :- Perfect Squres, 3, 4,, The lst digit of squre cnnot be, 3, 7, or 8 It cn only be 0,, 4, 5, 6, 9 Here is the list of the lst two digits of squre numbers: n In generl, we cll these numbers Squre These re the possibilities of squre numbers Notice tht the tens-digit of n odd squre number must be even The tens-digit of squre number ends in 6, must be odd The tens-digit of squre number ends in 4, must be even We cn mke use of these rules to deduce whether number is squre or not squre In ddition, here re the other properties of squre: Ech even squre cn be epressed s multiple of 4 (eg 4 k ) Ech odd squre cn be epressed s multiple of 4 plus (eg 4k ) 3 Ech odd squre cn be epressed s multiple of 8 plus (eg 8k ) 4 Ech 3n integer cnnot be squre 5 Ech 5n or 5n 3 integer cnnot be squre 6 Ech 8n, 8n 3, 8n 5, 8n 6, 8n 7 integer cnnot be squre 7 Ech 9n, 9n 3, 9n 5, 9n 6, 9n 8 integer cnnot be squre 8 If the tens digit nd ones digit of n integer re odd, it is not squre 9 If the tens digit is even nd ones digit is 6 of n integer, it is not squre Lstly, if nturl number N, such tht n N n, then N is not squre

58 -: 54 :- E A number is formed using 300 digits of nd some digit 0, cn it be rrnge into squre? E Prove tht there is no squre number in this number pttern:,,,, E 3 Find 4-digit squre number in the form of bb E 4 How mny squre numbers re there from to 000 inclusively? E 5 Given -digit number N, by dding nother digits on its left, becomes the squre of N Find ll such positive integers, N

59 Prctices -: 55 :- Prove tht for ny given nturl number n, n n is not squre Find the number of 4-digit number, fter dded 400, it becomes squre Prove tht m digits m digits is perfect squre 4 Find ll squre numbers with common lst 4 digits 5 Given tht nd y re positive integer, find ll the order pir of y, tht stisfy the eqution y 7 6 Given tht n is positive integer, find the smllest n so tht 008n is squre 7 Chin 99 If nd z re positive integers, prove tht the vles of z nd z 4 3cnnot be both perfect squres t the sme time 8 SMO Question 000 Evlte 000digits 000digits SMO Question 000 Let n be positive integer such tht n + 88 nd n 8 re both perfect squres Find ll possible vlues of n

60 Wht is Congruence? -: 56 :- Modulr Arithmetic CONGRUENCE Two integers re congruent modulo m if nd only if they hve the sme reminders fter division by m Let m be fied positive integer If,b,, we sy tht is congruent to b modulo m nd write b mod m m b whenever If m b, we write b mod m Another wy to look t Congruency The condition for to be congruent to b modulo m is equivlent to the condition tht b km for some k Z For emple, 7 3 mod 4, 6 4 mod 0, 73 mod, but 4 mod 3 0 mod 5 nd True or Flse? mod 7 ( ) 39 mod 3 85 ( ) 3 3 mod 7 ( ) 4 7 mod 3 0 ( ) Properties of congruency mod m (refleive property) (i) (ii) b mod m, then b mod m (symmetric property) (ii) b mod m nd b cmod m, then c mod m Cn you prove these properties? (trnsitive property)

61 If ' mod m nd b b' mod m -: 57 :- Modulr Arithmetic, then () b 'b' mod m () b 'b' mod m (3) b ' b' mod m (4)If c bc mod m nd gcd c, m, then it follows tht b mod m Cn you provide proof for these opertions? Congruences occur in everydy life The short hnd of clock indictes the hour modulo, while the long hnd indictes the minute modulo 60 For emple, 0 hours fter midnight, the clock indictes 8 o clock 0 8 mod In determining which dy of the week prticulr dte flls, we pply becuse congruence modulo 7 Two integers re congruent modulo if nd only if they hve the sme prity; tht is if nd only if they re both odd or both even The ide of congruence is not rdiclly different from divisibility, but its usefulness lies in its nottion, nd the fct tht congruence, with respect to fied modulus, hs mny of the properties of ordinry equlity Emple Wht is the reminder when 37 is divided by 7?

62 -: 58 :- Emple Wht is the reminder when is divided by 5? Emple 3 Find the reminder when is divided by 4 Tests for Divisibility Congruences cn be used to prove some of the fmilir tests for divisibility by certin integers The test of divisibility by 4 works becuse 0 mod 4 n is n integer nd, b re the lst two digits of this number, 00n b bmod 4 00 Given n integer 00 b, where n reminder when n00 b is divided by 4 is the sme s tht of b when divided by 4 Therefore the Emple 4 Prove tht number is divisible by 9 if nd only if the sum of its digits is divisible by 9

63 Emple 5 -: 59 :- Prove tht number is divisible by if nd only if the lternting sum of its digits is divisible by Fermt s Little Theorem If p is prime number tht does not divide the integer, then p mod p In ddition, for ny integer nd prime p, p mod p Eercise If nd b re divided by 5, it gives reminder nd 4 respectively Find the reminder when 3 b is divided by 5 If tody is Sturdy, wht is the dy of the week fter dys? 3 Find the reminder when is divided by 7 4 Prove tht mod Moscow 98 Find ll the positive integers n such tht n n is divisible by 3

64 -: 60 :- Lines nd Angles E Given tht AB : BC : CD : 3: 4 with E nd F being the mid points on AB nd CD respectively, if EF cm, find the length of AD A E B C F D E Given tht AB 0 cm on stright line, C is point on the sme line tht AC cm If M nd N re the midpoints on AB nd AC respectively, find the length of MN E 3 If Jenny is to trvel from point A to point B but need to rech the river bnk to wsh her hnds, wht is the shortest pth for her? *B A* River-- E 4 A wire of length 0 cm is to be bent to form pentgon, wht is the longest side of this shpe?

65 -: 6 :- E 5 For n nlog clock, if the time is from 030 to 050, how much does ech hour hnd nd minute hnd hve to move, in ngles? E 6 For n ngle A, if the rtio of its complementry ngle to its supplementry ngle is : 7, find the mesure of ngle A E 7 For n nlog clock strting t 5 O clock, how much will the hour hnd move before the minute hnd overlp it?

66 -: 6 :- Prctices For the given digrm, B nd C re two points on AD M nd N re the midpoints on AB nd CD respectively If MN nd BC b, find the length AD A M B C N D For the given digrm, A nd A 3 re two points on the line A A4 If A A 3 A A4, find the sum of length of ll the line segment between, A nd A 4 A, A3 A A A 3 A 4 3 Two points A nd B re on two sides of bo s show in the digrm, find the shortest line to drw from A to B ۰B A۰ 5 If the difference in two supplementry ngles is 8, find the complementry ngle of the smller ngle 6 From 6 O clock to 7 O clock, when will the hour nd minute hnds re t 90 prt? 7 From 4 O clock to 6 O clock, when will the hour nd minute hnds re t 0 prt?

67 -: 63 :- Prllel Lines E Given the digrm below, // b, line AB intersects nd b t A nd B IF CA bisects BA nd BC bisects ABb, prove tht C 90 A C B b Wht bout stright lines, b hve been intersected by nother stright line AB CA bisects BA nd BC bisects ABb If C 90, re the lines, b prllel to one nother? A C B b E Given the digrm below, AA // BA find the mesure of A B A A A B B A Wht bout this if AA// BA 3? A A B A B B A 3

68 -: 64 :- And how bout this if AA // BA n? A A B A B B B n- A n Question : If A B A 0, is AA // BA? A A B B A Question : If A A An B Bn, is AA // BA n? A A B A B B B n- A n E 3 For the given digrm below, AE // BD, 3, nd 5 Find C E D C A B

69 -: 65 :- E 4 Prove tht the sum of interior ngles of the qudrilterl is 360 Question : Wht bout the sum of ll interior ngles of polygon of n sides? E 5 For the digrm below,, D 90, EF CD, show tht 3 B E A 3 D F B C

70 -: 66 :- Prctices For the given digrm, AB // CD, GEF 90, B 00, EF bisects BEC, nd EG EF Find BEG nd DEG G B 00 F A D E C For the given digrm, CD bisects ACB If ACB 40, B 70 nd DE // BC, find EDC nd BDC A D E B C 3 For the given digrm, AB // CD, BAE 30, DCE 60, nd EF, EG trisects AEC eqully Are EF or EG prllel to AB? And why? A B F G E C D 7 Prove tht the sum of internl ngle of pentgon is 540

71 -: 67 :- 8 For the given digrm, CD bisects ACB, DE // AC nd CD // EF Prove tht EF bisects DEB A F D B E C

72 -: 68 :- Tringles E Show tht the sum of interior ngles of n-gon is given s 80n A A A n A 3 A 4 E Find the sum of ngles, A B C D E F A B C F D E E 3 Find the sum of ngles, A B C D E D A C B E

73 -: 69 :- E 4 In this ABC, the ngle bisector of B nd ngle bisector of D 30, find A ACE intersects t D If D A 30 B C E E 5 For the digrm below, A 0, ABC 90, ACB DCE, ADC EDF, CED FEG, find F F B D A C E G E 6 For the digrm ABC, BA etend to intersect the ngle bisector of ACE t D Show tht BAC B D A B C E

74 -: 70 :- Prctices For the given digrm, find the vlue of A B C D E A D C B E For the given digrm, find the vlue of A B C D E D A C B E 3 For the given digrm, find the vlue of A B C D E F D A F C B E 4 For the given digrm, find the vlue of A B C D E F G F A C D B G E

75 -: 7 :- 5 If the sum of ll interior ngles of polygon is s following respectively, find the number of sides of ech polygon () 60 (b) 60 6 Show tht the sum of ll eterior ngles of polygon is Chin 998 In right ngle tringle ABC, ACB = 90 o, E, F re on AB such tht AE = AC, BF = BC, find ECF in degrees 8 AHSME 996 Tringles ABC nd ABD re isosceles with AB = AC = BD, nd AC intersects BD t E If AC is perpendiculr to BD, find the vlue of C + D 9 SMO Question 995 In right ngled tringle, the lengths of the djcent sides re 550 nd 30 Wht is the length of the hypotenuse? 0 SMO Question 996 In the figure below, ABC, M nd N re points on AB nd AC respectively such tht AM : MB = : 3 nd AN : NC = 3: 5 Wht is the rtio of [MNC] : [ABC]? A M N B C

76 -: 7 :- SMO Question 998 In the figure below, AP is the bisector of BAC nd BP is perpendiculr to AP Also, K is the midpoint of BC Suppose tht AB = 8 cm nd AC = 6 cm Find the length of PK in cm A B P K C SMO Question 000 Determine the number of cute ngled tringles hving consecutive integer sides nd perimeter not more thn SMO Question 00 Two of the three ltitudes of right-ngled tringle re of lengths nd 5 Find the lrgest possible length of the third ltitude of the tringle

77 -: 73 :- Ares Are of tringle, S HB where H = height of tringle, B = the bse tht is to its height Are of prllelogrm, S HB Are of trpezium, S A B H where A nd B re the lengths of the prllel sides Some properties of re of tringles: () common bse nd common height of two tringles hve the sme re () the rtio of the res of two tringle is the rtio of the products of the bse nd height of ech tringle (3) the rtio of the res of two common bse tringles is the sme s the rtio of its heights (4) the rtio of the res of two common height tringles is the sme s the rtio of its bses E Given ABC, the three sides of, b, c corresponds to heights 4, 5, nd 3 Find the rtio of sides : b : c

78 -: 74 :- E Given the prllelogrm ABCD of re 64 squre cm, E nd F re midpoints of AB nd AD respectively Find the re of CEF D C F A E B E 3 In this ABC of re unit, BD DC, AF FD, CE EF, find the re of DEF A F B D E C E 4 In this ABC, E is mid point on AB, D is point on AC such tht AC: DC = : 3, nd BD nd CE intersect t F If the re of tringle ABC is 40 units, find the re of AEFD A E D F B C

79 -: 75 :- E 5 For the digrm below, ABCD is prllelogrm E nd F re point on AD nd AB respectively If BE DF nd BE nd DF intersects t O, prove tht C is equip-distnt from BE nd DF A E D F O B C

80 -: 76 :- Prctices For the given digrm, in this ABC, EF // BC, AE : EB m:, show tht AF : FC m: A D E B C For the given digrm, ABCD is trpezium AB // CD If the re of DCE is qurter of the re of DCB, find the frction of the re of DCE to tht of the re of ABD D C E A B 3 For the given digrm, in this ABC, P is point within this tringle AP, BP nd CP intersect on the opposite sides t D, E nd F respectively Given the re of 4 of the si smller tringles, find the re of ABC A B F 84 E 35 P D C

81 -: 77 :- 4 For the given digrm, P is point within this tringle Given tht the ltitudes from the vertices A, B nd C re h, h b nd h c respectively, nd the perpendiculr distnces from P to ech sides s t tb tc t, t b nd t c, show tht h h h A b c t c P t b B t C 5 For the given digrm, ABCD is trpezium BA nd CD produce intersection t O OE // DB, OF // AC E, B, C nd F re on stright line Show tht BE CF O A D E B C F 6 For the given digrm, P is mid-point on AC in the ABC PQ AC nd intersects t the AB produce t Q BR AC Show tht S ARQ S ABC C R P A B Q

82 -: 78 :- 7 SMO Question 998 In the figure below, tringle ABC is right-ngles tringle with B = 90 o Suppose tht BP AQ nd AC is prllel to RP If the re of tringle BSP is 4 squre nits, find the re CP CQ of tringle ABC in squre units A R S Q B C 8 SMO Question 999 In tringle ABC, D, E, F re points on the sides BC, AC nd AB respectively such tht BC = 4CD, AC = 5AE nd AB = 6BF Suppose the re of tringle ABC is 0 cm Wht is the re of tringle DEF? 9 SMO Question 004 In tringle ABC, AB = BC nd P lies within the tringle ABC such tht APB = BPC = CPA Given tht ABC = 60 o nd tht [ABC] = 70 cm, find [APC]

83 -: 79 :- Solving Diophntine Equtions Wht is Diophntine eqution? A Diophntine eqution is concerned with only integer solutions In some cses, it concerns only with positive integer solutions Wht mke solving Diophntine eqution interesting nd chllenging is tht there re more unknowns thn vilble equtions For emple, you my hve equtions but three unknowns The usul method of solving system of equtions will then need further development You will lso need to hve strong understnding in Number Properties So, enjoy the lesson E Jne spent five dollrs on number of ersers nd pencils An erser cost 30 cents, pencil cost one dollr How mny ersers nd pencils did Jne buy? E Find the integer solutions of 5 y 7

84 -: 80 :- E 3 Find the non-negtive integer solutions of 6y 90 E 4 Find the positive integer solutions of 79y 3 E 5 Find the integer solutions of 3707 y 5

85 -: 8 :- E 6 Find the number of wys of using numerous 5 or 7 only to sum up to ectly 4 E 7 Find the integer solutions to 9 4y 5z 000

86 -: 8 :- Prctices Find the integer solutions to: () 757 y, (b) 9y 44, (c) 039y 5 Find the positive integer solutions to: () 35y 9, (b) 5y 5, 3 Find the integer solutions to: () 5 8y 9z 50, (b) 39 4y 9z 78 4 Find the integer solutions to 5y 7z 3t 0 5 Find the positive integer solutions to 5 7y z 4 nd 3 y 4z 4 6 Chin m nd n re integers stisfying 3m + = 5n + 3 nd 30 < 3m + < 40 find the vle of mn 7 AHSME 99 If k is positive integer such tht the eqution in k? k 3k hs n integer root, then wht is 8 RSMO 983 Given tht pile of 00 smll weights hve totl weight of 500g, nd the weight of smll weight is g, 0g or 50g Find the number of ech kind of weights in the pile 9 Chin 00 How mn pirs of (, z) of two integers tht stisfy the eqtion z?

87 -: 83 :- Topicl revision () on Properties of Rtionl Number Given tht 5, b 3, nd b b, find the vlue of b Wht is the unit digit of ? 3 The numbers 49, 9, 9, 40,, 5, 53, 33, 3, 47 re grouped in pirs so tht the sum of ech pir is the sme Which number is pired with 5? 4 n! nn n 3 such tht n! is divisible by 990, is clled n fctoril Find the smllest positive integer n 5 Let A 63 nd B b5 be two 3 digit numbers If A B is divisible by 9, find the vlue of b 6 Find the sum of digits of Given tht, find the vlue of Find the vlue of such tht 3 is minimum SMO Wht is the lst two digits of 004 9? 0 The number A is formed by putting consecutive integers from 004 to 040 together Wht is the reminder when A is divided by 9? It is known tht n = hs 00 digit 9 How mny digits 9 re there in n 3? Find the number of digits in N where N is the product of ll the positive divisors of 00, 000, m nd n re two positive integers of reverse order (for emple 3 nd 3) such tht mn = Find the vle of m + n

88 -: 84 :- Answers: or or

89 -: 85 :- Topicl revision () on Integrl Algebric Epressions Evlute b 3 b, find the vlue of If 3 b 4b If b c, c 0, nd c b, find the lest vlue of b c 4 Find the sum of digits of the vlue of If, b, c re rtionl numbers, b c 0, nd bc 0, find the vlue of b c c b b c 6 If bc 0, b c 0, then when b c 00, find the vlue of 00 b c Given tht y 0, y 00, find the vlue of y If , then find the vlue of SMO Questions If is epressed s n integer, wht is the sum of its digit? 0 Find the ect vlue of If m m 3 0, wht is the smllest possible vlue of m 3 4m 004 0? Let nd z be two integers such tht > z Suppose tht + z = 0 nd z 38 Find z Find the sum of

90 -: 86 :- 4 Given tht, b, c nd d re rel numbers such tht b 3bc c vle of b c b b c c d d Find the lrgest possible Answers: c

91 -: 87 :- Topicl revision (3) on Solving Liner Equtions Find the number of different solutions of the eqution 3 Solve If 496 m m n, where m, n re integers, find the vlue of n 4 If there re infinite roots of for the eqution m 00 n m n 5 Solve the eqution 9, then find the vlue of k bk 6 For ll vlue of k, is lwys the solution of the eqution, find the 3 vlue of nd of b 7 Given tht the two equtions nd hs the sme 8 solution, find the vlue of 8 If 5 5, then find the vlue of 3 SMO Find the vle of the smllest positive integer m such tht the eqution ( m 5) (00m 9) 0hs only integer solutions