5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power


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1 Prim numbrs W giv spcial nams to numbrs dpnding on how many factors thy hav. A prim numbr has xactly two factors: itslf and 1. A composit numbr has mor than two factors. 1 is a spcial numbr nithr prim nor composit. Prim factors and factor trs A prim factor of a numbr is a factor that is also prim. Evry composit numbr can b xprssd as a product of prim numbrs. A factor tr is a good way to find ths prim factors. workd xampl 10 Writ 24 as a product of prim factors. Stps 1. Brak th numbr into a pair of factors (not using 1), thn brak ach subsqunt factor into a pair of factors. Circl any prim factors as thy appar, and stop th branch at this point. (Two possibl trs ar shown.) 2. Writ th numbr as a product of th circld factors. Solution = or Powrs Powrs ar a short way of writing rpatd factors. 5 2 is an xampl of a powr. Th small 2 is th indx and tlls us how many factors of 5 thr ar. Th 5 is calld th bas. For xampl: 4 2 = = = = bas Tutorial Tutorial 5 2 indx powr 1 whol NUMBERS 17
2 For 4 2 w say four squard or four to th powr of two or four to th scond powr. For 5 w say fiv cubd or fiv to th powr of thr or fiv to th third powr. For 7 4 w say svn to th powr of four or just svn to th fourth. Powr Man to bas can you har m? Anothr word for indx is xponnt. Numbrs writtn using indx notation ar in indx form or xponntial form. 4 is in indx form is in xpandd form. workd xampl 11 (a) Writ out 4 in xpandd form and work out th answr. (b) Writ out 7 4 in xpandd form and work out th answr. Stps Solutions (a) 1. Writ 4 in xpandd form. (a) 4 = Writ down th answr. = 64 (b) 1. Writ 7 4 in xpandd form. (b) 7 4 = Writ down th answr. = 2401 Powrs on calculators Many calculators hav a ky to hlp you work quickly with powrs, usually x y or ^. For xampl, if you want to work out 12, prss 1 2 x y = 5 x 2. To work out 5 2, simply prss. Tutorial Squars and squar roots A numbr is calld a prfct squar if it can b found by multiplying anothr numbr by itslf (i.. by squaring th othr numbr). Finding th squar root is th opposit to squaring a numbr. For xampl, to find th squar root of 16, you hav to think of a numbr that whn multiplid by itslf givs you 16. Th numbr is 4 sinc 4 4 = 16. This mans th squar root of 16 is 4. W writ this as 16 = 4. You can us your calculator to hlp with som calculations. Your calculator will hav a ky lik this:. To find 204, prss = or dpnding on th typ of calculator. Thn rad off th answr: 48. Tutorial Tutorial Tutorial 18 Hinmann MATHS ZONE
3 xrcis Prims, powrs and squar roots Prparation: Exrcis 1 Complt ths factor trs, thn writ ach numbr as a product of its prim factors. (a) 16 (b) = 50 = (c) 100 (d) 96 Worksht C = 96 = 2 Exprss ach numbr as a product of its prim factors. (a) 8 (b) 12 (c) 18 (d) 26 () 20 (f) 48 (g) 68 (h) 44 (i) 64 (j) 72 (k) 108 (l) 144 (m) 140 (n) 1000 (o) 800 (p) 400 Find thr diffrnt numbrs that can ach b xprssd as a product of four prim factors. (Th product may contain rpatd factors.) 4 Simplify ths with your calculator, using th shortst mthod possibl. (a) 21 2 (b) 18 2 (c) 48 2 (d) 61 2 () 75 2 (f) 88 2 (g) (h) 49 2 (i) (j) (k) (l) (m) (n) (o) (p) What numbr do you hav to squar to gt ach of ths? (a) 9 (b) 4 (c) 49 (d) 64 () 81 (f) 25 (g) 144 (h) Evaluat th following. (a) (c) () (g) (i) (k) (m) (o) 25 (b) 49 6 (d) (f) (h) (j) (l) (n) (p) Writ ach of ths in indx form. (Don t work out th answr.) (a) (b) (c) (d) What numbr tims itslf givs 25? Intractiv Qustions Qustions Worksht C1.9 Qustions Qustions 1 whol NUMBERS 19
4 () (f) (g) (h) (i) (j) (k) svntn cubd (l) thirtn to th powr of svn (m) ight to th fourth powr (n) nin to th sixth (o) lvn to th svnth (p) nin to th third 8 Writ ach of ths out in xpandd form, thn work out th answr. (a) 2 (b) 2 4 (c) 2 6 (d) 2 5 () 1 8 (f) 0 7 (g) 0 6 (h) 1 7 (i) 10 (j) 10 5 (k) 6 4 (l) 8 4 (m) 11 (n) 12 4 (o) 4 6 (p) What numbr do you hav to cub to gt ach of ths? (a) 1 (b) 27 (c) 8 (d) 64 () 8000 (f) 1000 (g) 125 (h) mans cub root. Th cub root of 27 is bcaus = 27. Evaluat, without using a calculator. (a) 8 (b) 64 (c) 1000 (d) 1 () 0 (f) 125 (g) 8000 (h) (a) Arrang ths numbrs in ascnding ordr (smallst to largst). 4 5, 5 4, 1 200, 10, 4 6, 5 5 (b) Arrang ths numbrs in dscnding ordr (largst to smallst) , 10 5, , 0 100, 2, 2 12 (a) Copy and complt: 10 1 = = = = = = 10 5 = (b) Th numbr was calld a googol by th mathmatician Edward Kasnr. (i) How many tims is 10 multiplid by itslf to gt a googol? (ii) Look at th pattrn in part (a). If you wr to work out what a googol is qual to, how many zros would follow th 1? (c) If you rais th numbr 10 to th powr of a googol, you gt a numbr calld a googolplx. (i) How many tims is 10 multiplid by itslf to gt a googolplx? (ii) If you wr to work out what a googolplx is qual to, how many zros would follow th 1? How much tim do you think you sav by writing a googolplx in indx form? Qustions Qustions hi.com.au 20 Hinmann MATHS ZONE
5 1 Without using your calculator, find btwn which two conscutiv whol numbrs ach of th following lis. (a) 10 (b) 5 (c) 20 (d) 62 () 99 (f) 2 (g) 70 (h) Without using your calculator, find btwn which two conscutiv whol numbrs ach of th following lis. (a) 10 (b) 52 (c) 1001 (d) 0 () 5 (f) 71 (g) 120 (h) 2 15 (a) Us your calculator to answr TRUE or FALSE to ach statmnt. (i) 4 6 is biggr than 6 4. (ii) 2 10 is biggr than (iii) 9 is biggr than 9. (iv) 19 2 is biggr than (b) Look at your answrs for part (a) and without using your calculator answr TRUE or FALSE to ach of ths statmnts. (i) 9 8 is biggr than 8 9. (ii) is biggr than Us your calculator to find th following through trial and rror. (a) A numbr that whn multiplid by itslf thr tims givs 4 (b) A numbr that whn multiplid by itslf thr tims givs 1728 (c) A numbr that whn multiplid by itslf thr tims givs 2744 (d) A numbr that whn multiplid by itslf thr tims givs 9 04 () A numbr that whn multiplid by itslf four tims givs 4096 (f) A numbr that whn multiplid by itslf fiv tims givs (a) Copy th following pattrn and complt it using your calculator = = = (b) Look at th pattrn in part (a) and without using a calculator copy and complt th following = = = Worksht C1.10 Homwork 1.2 Do ths in your had as quickly as you can and writ down th answrs $ $ What tim is it 5 hours and 40 minuts aftr 9.17 a.m.? of $ How many odd numbrs ar thr btwn 20 and 6? 10 Find th missing numbr: 816, 408,, 102. Tim targt: 2 minuts 1 whol NUMBERS 21
6 Prim numbrs vrsus trrorism Trrorism has cost many innocnt livs so far this cntury, and mathmatics has a rol to play in stopping trrorist acts bfor thy happn. Trrorists ar abl to commit ths acts bcaus thy can plan and communicat in scrt. Govrnmnt intllignc agncis all ovr th world try to intrcpt trrorist computr mails and dcod thm. At th tim of th trrorist attacks in 2001, ncryption or coding tchniqus xistd that couldn t b crackd. Ths ncryption tchniqus ar basd on prim numbrs. All ncryption is basd on a ky, which is a word or a numbr. Ths kys giv th information ndd to dcod or dcrypt th mssag. Th kys currntly most commonly usd to ncrypt mails ar numbrs that ar th products of two prim numbrs. To crack th cod, you hav to work out th original two prim numbrs. Whn th numbrs involvd ar vry larg, this is almost impossibl to do bcaus thr ar so many possibilitis. Th largst prim vr found has ovr 2 million digits in it, and thr ar always nw prims bing found. Evn th most powrful computrs in th world would tak cnturis to find th two prims involvd in ncryption. Multiplying two prims is an ffctiv ky for ncryption bcaus it is asy to do in on dirction (multiplication), but xtrmly difficult to do in th othr dirction (finding factors). Govrnmnts ar hoping that in th futur quantum computrs that can procss calculations simultanously will b abl to find th two prim numbrs quickly and nabl thm to crack th trrorists cods. Braking an ncryption cod is calld dcryption. 22 Hinmann MATHS ZONE
7 Qustions 1 Which two prim numbrs wr multiplid togthr to giv ach of ths ncryption kys? (a) 77 (b) 8 (c) 65 (d) 202 () 14 (f) Multiply th following prim numbrs togthr to gt ncryption kys. (a) and 17 (b) 7 and 1 (c) 47 and 7 (d) 101 and 10 () 11 and 727 (f) 1 and 9 19 (a) Writ out th first fiv prim numbrs. (b) Writ out th six smallst kys that you can crat by multiplying two prims. (c) How many factors dos ach ky hav? (d) Do you think vry ky cratd by multiplying two prims togthr will hav this numbr of factors? Why? 4 Hr s a simplifid vrsion of how dcryption using prim numbrs works. Suppos th ncryptd mssag is W L R G Y S B E M R K C R E B J C V and you know th ky is 71. You hav to find th two prim numbrs that multiply togthr to giv 71. This is rasonably asy for such a small numbr. Th answr is 2 and 1. Ths digits tll you how much th lttrs in th original mssag hav bn shiftd backwards along th alphabt to giv th codd mssag.you match ths digits to th cod as follows: W L R G Y S B E M R K C R E B J C V Th way dcrypting this cod works is that th numbr tlls you how many lttrs to shift ach lttr forwards along th alphabt. Th 2 undr th W mans shift two lttrs forward through th alphabt from W, so th uncodd mssag, usually calld th plaintxt, has Y as its first lttr. For th scond lttr you find th lttr thr placs on from L in th alphabt, which givs O as th scond lttr of th plaintxt. (a) Continu dcrypting th cod to gt th ntir original plaintxt mssag. (b) Dscrib what you did with Y 2. (c) Why couldn t you us any numbr (.g. 120) as a ky? 5 Dcod this mssag givn th ky 94. K X P G Q F O N D M N H K B E L N L N S Y K Z D Not that you nd to work out which ordr to put th two prims in. Only on will giv you th mssag. 6 Encrypt your own mssag using th prim numbr systm. Us only prim numbrs undr 100 and kp th mssag undr 25 lttrs. Giv your codd mssag and ky to somon to dcrypt. Rsarch hi.com.au Find out about th GIMPS (Grat Intrnt Mrsnn Prim Sarch) projct and prsnt a rport on how prim numbrs ar usd in mail ncryption. Discuss whthr govrnmnts should ban ncryption and mak it possibl for intllignc agncis to rad all mails. 1 whol NUMBERS 2
8 Simplify th following with th hlp of a calculator, and brak th cod to find th caption. R W C C O (8 + 5 ) 16 T 4 10 A O L A 6 5 U Hinmann MATHS ZONE
9 Summary St out solutions to wordd problms by putting hadings and writing calculations in logical squnc. Estimat products and quotints by rounding numbrs to th lading digit. Ordr of oprations rquirs bracktd calculations to b don first, working from innr brackts, thn multiplication and division as thy appar, thn addition and subtraction as thy appar. Prims ar numbrs with xactly 2 factors: thmslvs and 1. Indx notation is a way of writing rpatd factors as a powr of th factor. Taking a squar root is th opposit of squaring a numbr. FAQs What is th largst multipl of 9? Numbrs don t hav largst multipls. Th multipls of 9 continu forvr: 9, 18, 27, 6, Is 1 a prim numbr? No. A prim numbr has two factors. 1 is nithr composit nor prim. Skills 1 Us rounding to th first digit to stimat ths products. (a) (b) (c) Us rounding to th first digit to stimat ths quotints. (a) (b) (c) Us rounding to th first digit to stimat ths, and thn us a calculator to work out how far off your stimat was from th xact answr. (a) (b) (c) Work ths out using your calculator. (a) 258 (21 162) (b) [(82 94) 41] Find, without using a calculator: (a) 9 (2 + 1) 2 (b) (c) (1 5 2) + (20 10) (d) [5 (9 + 1)] 6 Choos th corrct answr. In th calculation of 2 [0 (4 1)] + 6 th first opration to do is: A + B C D E any of ths symbols 7 Choos th corrct answr. (5 + 6) 2 + (15 2) 6 is qual to: A 40 B 20 C 2 D 25 E , whol NUMBERS 25
10 8 Find th first thr multipls of: (a) 7 (b) 10 (c) 12 (d) 52 9 Find th LCM of: (a) and 8 (b) 4 and 16 (c) 6 and 9 (d) 20 and Choos th corrct answr. Th LCM of 2, 5 and 9 is: A 10 B 18 C 45 D 90 E Find all th factors of: (a) 5 (b) 1 (c) 44 (d) 48 () 51 (f) Find th HCF for ach of ths. (a) 11 and 55 (b) 90 and 60 (c) 72 and 64 1 Choos th corrct answr. Th HCF of 14, 28 and 56 is: A 14 B 1 C 7 D 56 E Writ factor trs and xprss ach numbr as a product of prim factors. (a) 24 (b) 0 (c) 88 (d) Writ ach of ths numbrs in indx form. (a) (b) tn cubd (c) fiv squard (d) 12 to th powr of 8 16 Writ ach of ths out in xpandd form, thn work out th answr. (a) 5 (b) 8 4 (c) ( 2 2 ) Evaluat th following. (a) 64 (b) 900 (c) 225 Applications 18 Put brackts into ths statmnts, whr ncssary, to mak thm tru. (a) = (b) = 7 19 (a) Copy th following pattrn and complt it using your calculator = = = (b) Look at th pattrn in part (a) and without using a calculator copy and complt th following = = = 20 (a) Look at this pattrn: 2 2 = Now copy and complt ths. (i) 2 = (ii) 4 2 = 2 + (iii) 5 2 = (b) Dscrib th rlationship btwn a squar numbr and th squar numbr bfor it. (c) Using th rlationship stablishd in part (b), complt ths. (i) 12 2 = (ii) 20 2 = Is 2 10 largr or smallr than 1000? Writ th diffrnc. 1.,, 26 Hinmann MATHS ZONE
11 22 Put ths numbrs in ascnding ordr: 2 4, 121, 10 2,, Writ ach numbr as a product of its prim factors, in indx form. (a) 81 (b) 128 (c) 144 (d) Simplify ths with your calculator, using th shortst mthod possibl. (a) 6 5 (b) 1 6 (c) 15 (d) 25 () 2 (f) 2 7 (g) 6 (h) 4 5 (i) (j) (k) (l) 6 5 (m) (n) (o) (p) St out ths calculations in your normal way and work out th answrs. (a) (b) (c) List th numbrs if you count by nins, starting at 0 and nding at 75. Copy and complt th following by writing < or > btwn th numbrs. (a) (b).9.8 (c) Copy and complt ach of th following by finding th pattrn. (a) 9, 15, 22, 0,,, (b) 1,, 7, 15,,, 5 Calculat: (a) (b) (c) Find th missing numbr that maks ach of th following tru. (a) + 7 = 24 (b) 4 = 15 (c) 7 0 = 15 7 Prform th following divisions. (a) 768 (b) (c) List all numbrs that 8 gos into that ar gratr than 70 and lss than Simplify: (a) (b) 18 6 (c) 8 (15 5) 10 Which whol numbr is ach of th following dcimals closst to? (a) 1.9 (b) (c) Calculat: (a) (b) (c) If Cln purchass thr tops at $15.50 ach, how much chang will sh rciv from a $50 not? 4 6 Worksht R1.7 Worksht R1.8 Worksht R1.9 Worksht R1.10 Worksht R1.11 Worksht R1.12 Worksht R1.1 Worksht R1.14 Worksht R1.15 Worksht R1.16 Worksht R1.17 Worksht R1.18 Worksht C1.11 Worksht C1.12 Assignmnt 1 1 whol NUMBERS 27
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