Shuli s Math Problem Solving Column


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1 Shuli s Mth Problem Solvig Colum Volume, Issue Jue, 9 Edited d Authored by Shuli Sog Colordo Sprigs, Colordo Cotets Mth Trick: Metl Clcultio: b cd Mth Competitio Skill: Divisibility by 3 A Problem from Rel Mth Competitio 4 Aswers to All Prctice Problems i Lst Issue 5 Solutios to Cretive Thikig Problems 55 to 57 6 Clues to Cretive Thikig Problems 58 to 6 7 Cretive Thikig Problems 6 to 63 Mth Trick The Trick I the lst issue we preseted how to clculte the followig multiplictios metlly: These multiplictios re i the geerl form: b cd where, b, c, d d re digits with b close to cd I this short lesso I will preset other short cut through exmples Exmple Metl Clcultio: b cd Clculte 8 3 If we tret 8 s with multiplictio i b Step : Clculte b or b I this exmple, 8 3, the this is or 3 Step : Multiply the result i step by I this exmple, 4 Step 3: Clculte b I this exmple, 3 6 Step 3 Add them this wy: We re doe: Exmple Clculte 4 3 Tret 4 s 3 with 6 The this is multiplictio i 3 3b Step : Clculte 3 b or 3 b I this exmple, 4 6 or Step : Multiply the result i step by 3 I this exmple, Step 3: Clculte b I this exmple, 6 Step 4 Add them: We obti Exmple 3 Clculte 47 5 Tret 47 s 5 with 3 The this is multiplictio i 5 5b Step : Clculte 5 b or 5 b I this exmple, or Step : Multiply the result i step by 5 I this exmple, Step 3: Clculte b I this exmple, 3 3
2 Shuli s Mth Problem Solvig Colum Volume, Issue Jue, 9 Step 4 Add them: We obti Exmple 4 Clculte Tret it s multiplictio i 7 7b Step : Clculte 7 b or 7 b I this exmple, Step : Multiply the result i step by 7 I this exmple, Step 3: Clculte b I this exmple, 4 8 Step 4 Add them: or We get Prctice Problems Mth Competitio Skill Defiitios: Divisibility by Alterte Digit Differece The st, 3 rd, 5 th, digits couted from the right re clled odd plced digits The sum of ll these digits is clled the odd plced digit sum The d, 4 th, 6 th, digits re clled eve plced digits The sum of ll these digits is clled the eve plced digit sum Subtrct the eve plced digit sum from the odd plced digit sum The result is clled the lterte digit differece Exmple Wht is the lterte digit differece of 378? Aswer: The odd plced digit sum is The eve plced digit sum is 3 5 The the lterte digit differece is 5 5 Exmple Wht is the lterte digit differece of ? Aswer: 5 The odd plced digit sum is The eve plced digit sum is The the lterte digit differece is 5 5 Divisibility by We hve the followig theorem for divisibility by Theorem A umber is divisible by if d oly if the lterte digit differece of the umber is divisible by Note tht is divisible by y turl umber Exmple 3 Is 47,839 divisible? Aswer: Yes The odd plced digit sum is 9 8 4, d the eve plced digit sum is 3 7 The the lterte digit differece is, which is divisible by So 47,839 is divisible by Exmple 4 Is divisible? Aswer: No The odd plced digit sum is , d the eve plced digit sum is The the lterte digit differece is 5 5, which is ot divisible by So is ot divisible by Exmple 5 Is 789 divisible? Aswer: Yes The odd plced digit sum is 4, d the eve plced digit sum is The the lterte digit differece is 4 6 which is divisible by So 789 is divisible by Proof of the Theorem Let N be digit umber is odd without loss of geerlity Assume Express N i the bse expsio: ' s ' s 99 Copyright 9 Shuli Sog All Rights Reserved Use with Permissio
3 Shuli s Mth Problem Solvig Colum Volume, Issue Jue, 9 m ' s 9' Note tht d 999 m s 9 where m is eve re lwys divisible by Therefore, N is divisible by if d oly if 3, which is the lterte digit differece 3 3, is divisible by Remider upo Divisio by For umber, clculte the lterte digit differece If the result is more th or equl to, clculte the lterte digit differece gi Or subtrct from it If the result is still more th or equl to, subtrct gi util the result is less th If the lterte digit differece is less th, dd to it If the result is still less th, dd gi util the result is lrger th or equl to The fil result is the remider of the umber upo divisio by Exmple 6 Wht is the remider of upo divisio by? Aswer: 6 From exmple, the lterte digit differece is The 6 is the remider Exmple 7 Wht is the remider of upo divisio by? Aswer: The odd plced digit sum is , d the eve plced digit sum is 3 6 The lterte digit differece is The is the swer Exmple 8 Sixdigit umber Aswer: 8 Exmples of Problem Solvig 7m 3 is divisible by Fid m The odd plced digit sum is 3 5, d the eve plced digit sum is m 7 8 m The lterte digit differece is 5 8 m 3 m The oly vlue of digit m is 8 such tht 3 m is divisible by Exmple 9 _ Fivedigit umber m497 is divisible by where m d re digits with m How my differet pirs of vlues for m d re there? Aswer: 8 The odd plced digit sum is m 6, d the eve plced digit sum is 4 The lterte digit differece is m The m is divisible by So m m could be from to 8, d from to 9 respectively Therefore, there re 8 differet pirs of vlues for m d Exmple! b where d b re digits Fid d b Aswer: d b 4! is divisible by 9 The the digit sum b 48 is divisible by 9 So b 6 or b 5! is divisible by The odd plced digit sum is 5, d the eve plced digit sum is 3 b The lterte digit differece is b The b is divisible by So b or b 9 Solvig for d b we hve d b 4 Exmple Prove tht y sixdigit umber bc, bc is divisible by where, b, d c re digits with Proof: The odd plced digit sum is c b, d the eve plced digit sum is b c The lterte digit differece is, which is divisible by Therefore, bc,bc is divisible by Prctice Problems Circle the umbers divisible by : 34 3, Fid the remider for ech upo divisio by : Is divisible by? 4 Whe is divided by, wht is the remider? 5 Eightdigit umber Fid m m 37 is divisible by 6 9! b where d b re digits Fid d b _ 7 Fivedigit umber m48 is divisible by where m d re digits How my differet pirs of vlues for m d re there? _ 8 Prove tht y sixdigit umber bccb is divisible by where, b, d c re digits with Copyright 9 Shuli Sog All Rights Reserved Use with Permissio 3
4 Shuli s Mth Problem Solvig Colum Volume, Issue Jue, 9 A Problem from Rel Mth Competitio Tody s problem comes from MthCouts The problem or similr problem ppered i MthCouts d other mth competitios occsiolly (MthCouts 995 Ntiol Sprit Problem 7) A chord of the lrger of two cocetric circles is tget to the smller circle d mesure 8 iches Fid the umber of squre iches i the re of the shded regio 8 Prctice Problems (MthCout 7 Ntiol Trget Problem 7) Two cocetric circles with rdii of 9 d 9 uits boud shded regio A third circle will be drw with re equl to tht the shded re Wht must the rdius of the third circle be? Express your swer i simplest rdicl form 9 9 Aswer: 8 Theorem I the figure below, C d C re two cocetric circles Chord AB of circle C is tget to circle C Let l be the legth of AB The the re of the rig betwee the two circles re solely determied by l, idepedet of the sizes of the two circles, provided tht l is fixed B l A C (5th AMCB 4 Problem d 55 th AMC B 4 Problem ) A ulus is the regio betwee two cocetric circles The cocetric circles i the figure hve rdii b d c Let OX be rdius of the lrger circle, let XZ be tget to the smller circle t Z, d let OY be the rdius of the lrger circle tht cotis Z Let XZ, y YZ, d e XY Wht is the re of the ulus? Y d Z O c e b X C Proof of the Theorem: Let T be the tget poit, d O be the ceter of the two circles Let R d r be the rdii of circles C d C respectively Drw OA d OT The OT AB d T is the midpoit l of AB Obviously, AT, OT r d OA R I right OTA B l r 4 R So R l l The the re of the rig is R r 4 determied by l solely l C r T O r 4, which is With l 8 the swer to the problem is R A C A) B) b C) c D) 3 (th AHSME 969 Problem 6) d E) e The re of the rig betwee two cocetric circles is squre iches The legth of chord of the lrger circle tget to the smller circle, i iches, is A) 5 B) 5 C) 5 D) E) 4 (6th AMCA 9 Problem 9) Adre iscribed circle iside regulr petgo, circumscribed circle roud the petgo, d clculted the re of the regio betwee the two circles Bethy did the sme with regulr heptgo (7 sides) The res of the two regios were A d B, respectively Ech polygo hd side legth of Which of the followig is true? 5 5 A) A B B) A B C) A B D) A B E) A B 5 5 Copyright 9 Shuli Sog All Rights Reserved Use with Permissio 4
5 Shuli s Mth Problem Solvig Colum Volume, Issue Jue, 9 Aswers to All Prctice Problems i Lst Issue Metl Clcultio Divisibility by 9 The umbers divisible by 9 re: 34 33, Yes A Problem from Rel Mth Competitio 6 divisio by 8 So it is i the leftmost colum Therefore, cot be the sum If is the sum of the ie umbers i mtrix, the cetrl umber must be Sice mod 8, the cetrl umber is t the third colum from the left Therefore, c be the sum of the ie umbers i mtrix 56 Coverig with Tetromios You my hve got o s the swer if you hve tried It is correct But re you ble to expli why you cot? I gve the problem to child oce He hd tried little while, d the he sid I kow the swer is o, d I feel tht the Tshped is strge Yes, the Tshped is the odd m Why is it? Color the bord with the stdrd chessbord colorig: Solutios to Cretive Thikig Problems 55 to x 3 Mtrix Look t the mtrix If we pir the umbers s show, we kow tht the sum is 9 times the cetrl umber The y oe of the followig four tetromioes covers two blck d two white squres: Therefore, the sum must be multiple of 9 Two umbers i ech pir hs sum of 58 = x 9 95 is ot multiple of 9 So the sum cot be 95 However, if umber is multiple of 9, it is ot ecessrily the sum For exmple, 44 is multiple of 9, but it cot be the sum of the ie umbers i mtrix The reso follows If 44 is the sum of the ie umbers i mtrix, the 44 cetrl umber must be 6 However, 6 is i the 9 rightmost colum If umber is i the leftmost colum, it cot be the cetrl umber of mtrix either Both d re multiples of 9 If is the sum of the ie umbers i mtrix, the cetrl umber must be Recll the shortcut for the divisibility by 8 i Issue 6, Volume yields remider of upo There re blck d white squres Wherever you plce these four shpes o the 4 5 rectgle, two ( 4 ) blck d two white squres will be left ucovered These four ucovered squres (two blck d two white) cot be covered by becuse the Tshped tetromio covers three blck d oe white squres, or oe blck d three white squres oly 57 Weighig Met II Agi, let us study from smll umbers We must hve weight of poud for piece of met of poud As we tlked i the Weigh Met I problem (Issue 5, Volume ), we do t wt to mke other weight of poud to weigh piece of met of pouds Isted we would like to mke hevier weight A weight of pouds works Copyright 9 Shuli Sog All Rights Reserved Use with Permissio 5
6 Shuli s Mth Problem Solvig Colum Volume, Issue Jue, 9 However, we my plce piece of met d weights o oe p If we plce piece of met of pouds with the existig weight of poud together, we would like to mke weight of 3 pouds to blce them With the weight of 3 pouds, we c weigh piece of met of 3 pouds We c weigh piece of met of 4 pouds by combiig the two existig weights To weigh piece of met of 5 pouds, we eed ew weight Sice we my plce the met of 5 pouds with the two existig weights ( poud d 3 pouds) together o oe p, we would like to mke weight of 9 pouds to blce them The we c use the 9poud weight to blce piece of met of 6 pouds d the 3poud weight Combie the 9poud weight d the poud weight to blce piece of met of 7 pouds d the 3poud weight Use the 9poud weight to blce piece of met of 8 pouds d the poud weight Usig the 9poud weight we c weigh piece of met of 9 pouds Combiig the 9poud weight d the poud weight we c weigh piece of met of pouds Combie the 9poud weight d the 3poud weight to blce piece of met of pouds d the poud weight Combiig the 9poud weight d the 3poud weight we c weigh piece of met of pouds Usig ll three existig weights we c weigh piece of met of 3 pouds To weigh piece of met of 4 pouds, we eed ew weight Similrly we would mke weight of 7 pouds to blce the met of 4 pouds d ll three existig weights (poud, 3poud, d 9poud) Now we hve the ptter: the weights re powers of 3 So the fifth weight is 8 pouds Five weights re eough, which re poud, 3 pouds, 9 pouds, 7 pouds, d 8 pouds respectively I fct, with these five weights we c weigh piece of met of up to pouds To weigh piece of met of up to pouds, you my hve differet set of five weights Oe set my be five weights of poud, 3 pouds, 9 pouds, 7 pouds, d 6 pouds respectively Clues to Cretive Thikig Problems 58 to 6 58 A Divisio Fctorize Movig Checkers Let me tell you the first move: 6 A Checkerbord Gme For gme strtegy, workig bckwrds very ofte helps Cretive Thikig Problems 6 to 63 6 Four Tromios to Oe Arrge the four Lshped tromios together to mke lrger shpe similr to them 6 Mgic Circles Fill to ito the smll circles such tht the five umbers o ech of the three lrge circles hve sum of 4 Which umber must replce the questio mrk? 63 Two Smrt Studets of Dr Mth Dr Mth hs two smrt studets Al d Bob Dr Mth picks up two itegers m d from to 9 with m Dr Mth clcultes the sum of m d d tells Al the sum, d clcultes the product of m d d tells Bob the product The Al d Bob hve the followig coverstios: Al: I do t kow wht m d re, but I m sure you do t kow either Bob: Now I kow wht m d re Wht does Dr Mth tell Al? First move (Clues d solutios will be give i the ext issues)? Copyright 9 Shuli Sog All Rights Reserved Use with Permissio 6
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