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midterm-sols

# midterm-sols - Math 115A(Santos Name g Mid-term exam...

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Unformatted text preview: Math 115A (Santos) Name: g Mid-term exam Student ID#: ' V November 2nd 2007 INSTRUCTIONS (1) READ INSTRUCTIONS CAREFULLY!!! (2) DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO DO SO (3) FILL IN THE INFORMATION ON THIS PAGE (your name, etc) NOW!! (4) SHOW YOUR WORK on every problem. Correct answers with no support work will not receive full credit. (5) PLEASE WRITE LEGIBLY. Be organized and use the notation appropiately. (6) NO EXTRA ASSISTANCE ALLOWED. Assistance from classmates, notes, books or calculators is prohibited. You should only have a writing and an erasing implement on your desk. (7) YOU MAY USE THE BACK SIDE OF THE PAPER. # Student’s Score Maximum possible Score 1 10 2 8 3 l‘ 10 _" .4 8 Total points 36 _J Note: all the numbers a, b, . . . ,n, . . . ,zr, . . . in this exam are assumed to be integers. 1. (10 points) Use the properties of gcd’s (and not the Fundamental Theorem of Arithmetics) to prove the following: a) If p is a prime and p divides ab but does not divide a, then p divides b. b) If club, then c|(a,c)(b,c). C) “ab E 0 (mod m) and m is prime” implies “a E 0 (mod m) or b E 0 (mod m)”. The K17 fl“? Fey 5’ he eraéeg/ [‘4 W W C; {Laval 0“ W WVWIK (Alb): Maw/1E . 1”an Has 09 515mm P Obie ‘M‘l BL‘VEOLQ a, M F i; Fawn/(7 we law/ﬂ 6P1“) : j ﬂwm ~' /: Px+ay Z? 5: bpxi— EQX/ toll/{CL f." at JAVA: i7 l) sail/ice, f) lalp _. 10> WVLLC (Ox/C) (L216): CA)“, 57/) (bm+c V1) : - *2 r; alpxww— O‘Cqur lac/WI + C7”? TIA/3; 5; 0A W/sz’ZZL ‘g g c, 3,04% C /o\l> a) 77W haul-£0) [Mylo ”Olivia/Ekaczilyé wlml 70” Hero! d2, PW ”M W” a, WW 0 PM“: w a or MIL * J is wlm We l/uxve Fl’vL'idt‘h (at)! 2. (8 points) Show that the following numbers are irrational. (Warning: none of them is a root of a monic polynomial. Clue: in both cases you can easily derive that they are not rational from the Fundamental Theorem of Arithmetics): a) 3/2 b) log23. V 09 If l/E/z; 5) Val-{owj/ 6,! /3/2 :- q/b W 5:. :~ :1 ,. ov‘ gy; 20g Tn: wilwaclf "H’ké f7 l4 3 ‘Lh yllw 3104i; V‘L'}(jt LEO L1 (‘2? Z 19 YLILQ JQCLY ”Zn“ Wﬁl “Fla/“M wEJ’lﬂ :va/i QXFQH—é/I/LL/ \x/lA/Clt in Ha jacloviyk ‘Qn i 20») g)» €«FWJ wind mu WOW; A l) [20313-5 % :> 3:2 /”;—> 35:2q which 43):“ comlrvxaacll; WW (#‘C'Al’im/‘io‘l. 3. (10 points) Find a solution and write the general form of all solutions to the following Diophantine equation. 416cc +1003] =12. l; Y5“ i1 * ‘— Q- 5')” 1 cl W m7 M 9 l 0 {6’69 ”/0” 5/ g 0 / M L/ 0 L/ 4 - L/ E ’5 25 (“/0 libL 3C6! M g: “gxlrlfg 425,;4‘90 =5?) 127’7ﬂxq4{+Z§g/@O [5 came mlmlﬂoi1_ Wm game/raj gob/”lieu [s N‘\ 1) if— . _... _..,,. A X 7 )(o + 1/1 :- ge / g + ij (a, b) L_ ....... __ 7: V0“ (gig) “ 3M League in WWW aka ml yolk 30;” / . how/m: iivé ﬁlm/[tow .M i E; oLviou/g +LW\2’ it . M [email protected]"[email protected] q/K M /C’@ W 4. (8 points) For each a E {O,1,...,9}: a) If a has an inverse modulo 10, ﬁnd it. b) Say how many solutions does the linear congruence am E 4 (mod 10) have. 0) 4! ET“??? 65-4%»sz Q L94 CWFffKK® (ox/0) :/ (w. asz/Wa/ ,4; ML m @925 be} 4 1‘; m1) AZﬁzch/QVL 4n [mW'Je (26 CL Walt/Ll: -/§ :5 a @Mm X 9WD Hm} “Ana :5 (2&7 MM ax : /+ {07 . E; H/xL Pwriekl’y if Lng/ vim exggvz; 2/4”“! {mp7 6 [(5919): 1 - , Lg; M Log/1; dgm/ 41% {MW 330 4 3; 7 Wm! 77m}? wa/errzor wa)’ Hug‘ : /./s/[WQ//a)/ MsﬂmJ/cz) W 7975/ [We/my b) [Va-W, Oxle] (Mpg/[0) 2494 41 ’ggLI/LJ’LOM gem/Ml on} g L, ifék mM/ptzfpi—L 0; VIM ng' (% QQ'WJ/lg‘ , m y Hm Wwogfgo’A/LLCWA a ‘37 (a, {(3)ng mow/ ‘ arS *9 Cams; w» m emigre“ &:-2,Ll,é,§ "‘3’ (CK, {(3)52 ““> ILW‘CJ’ sch‘leow &:{,'3/ 7,“). --> (rq,/0):/ -—> one schl'Cam‘ a : o .4? WC 94,9):ié9. ~> mayhem ‘24“ 94 mum! #0 Maul/M ;;;«_,n+ m7 “CL/‘0‘] ML/lzwoué guy-e. :. W — w _ "-1? :r 9 / mgcwhaw‘ ar3+x~3 at”; JC :9qu g/ C“:/ ﬁx:‘-{ / 01:9r~>1(:’/or( a: ”5x, 1111 Q33 ax:gorg a: Z» erar710L35~>no Sch/0M ’ N ...
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