Midterm1Solns

# Midterm1Solns - Midterm 1 Name Math 310 ~ Section 3 Fan...

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Unformatted text preview: Midterm 1 Name: Math 310 ~ Section 3., Fan: 2010 E '1 ‘ Q; a! I Problem Points Your SCOTS . .. mi it) 15 5 IS? . a 20 (12:2 E15 w I 10 {a 10 1 t .10 it) i100 g 3&6 E CLO 01 “1CD Oi} E FF] or c+ gang-J By signing below, I certify that I am striving to adhere to the principies of academic truth and integrity held by B-‘Iichigem State U11'wez'Sit}!E and forth in. the Spartan Life Student Haﬂ<ﬂ1xmk Signature: Math 310 — Sectiim 3. Fail, 201.0 Aficiimm "J 1. [.10 paw} Complete the deﬁnitimiis of the. terms in bold: a. Let a, and {a be. mmgm'a witi‘1 b ,i U. We say \$136er divides a if . . . m C "g‘W- §ame I ygn'lﬁ’gﬁ-Ef y in. Let: a, and w he integers with n > 0. Tim congruence class of a modulo n dammed {a}; is ' 6; “3% ".éthgﬂ gﬂ‘éﬁfgm can; {:¢5!§F0391‘%“ F \$0 -- magi-ﬁx; p1,, r yam, Eb g M12! m3 £3 3% {Wm} ﬁx 2. {10 13%.] Shaw by exalm‘ﬂe that if {H}. + 121' : d > 14 £11101} (3.. 32) Elm}? not be (i. {mg-win" €13; E 3 b SW 6 "m {4,1,} + \$1 if) =-= 2 ‘:> "21} m" {3? 55‘} \$3., \$52. Math 330 w Section 3. Fail, 13010 Aﬁdtczrm IE 3. {15 ptsi Let. n he a positive integer. Prove that, a, and r: ieave the SEMIM.‘ rcmaimier when divided by n if anti only if a c. I HR: for 301110 integci' A7. [haughty] gm DWDEW ﬁgﬁﬁ/EWWV’" '{0 a; ﬁnal a; “354% 9 O§m<n O Erl 4:“ Ecrha “Qil'ii'r‘, {"1312 (27 an / 3W2; n in <1“ ~— y"; {4: O} Mvﬁiﬂiha fk wgmd 5+HW.¢¥ hszh*ﬁs 5? “h <ﬂ”% < h& Gum} aﬁaizn} g"? '{T'E’J‘gq Saga-“g hm???“- camc; :- (ngym‘lqﬁﬂgﬁm "‘ “(ix‘“‘i5;\) 7'1 ﬂ “692.3 4 + U} “ {90) {thafya‘i‘f} ‘KT: CﬂmC '1 he? 199"" Shane. (1 HM m 1‘: M“) whiﬁw‘l“ 3? hﬁwl ~E~£r1~fgjl A (r “{1} 5: n (ié'mﬁw *§13ﬂ Eu+ wn*irs“*a5“; M “ECH"V2)E? % 4.. gm pits} Prove that over}; integer n, :> 1 ii; the product of primes. (Hint: Let S be the act of AH integE-ars greater than 1 that are not the jin'mtluct 62f 1:}1‘111195. U a proof by coxltradicti()n to shrinw that 8 must be empty.) Lair in «m QE‘ f5 5*- \$5,! (1 M" «:2 i m Wag in * Ofﬁiév'ﬂiﬁ‘? X'Eamr M Til-£3 Left/\$1? {1“} 5": imagiéj‘f QIﬁmg-ij/ r5 '2 Math 3310 — Section. 3. Fall, 20H} Miﬁierm 1 3:). [15 pm] a. Prove that if 2 3'11 Z71,“ than ((1.713) = if... m: {3%- J b. Show by e‘xmnphw that evm if" (can) Cram ﬂail-1N” (:3. [1%] mg Th Tkm m 1-131 "Hm ﬁlmy mm {54- W?" W} + “M i» "on W» .. ‘€QE'&+E\$% MMMM (“WWW Mffv‘méﬁ {1 “NE m 7% U m 5% Ha.er mil???“ {n‘irégﬂlw wkﬁak {m 3m Eg'gfﬁjfegf a} £3 Eg‘qmr mmég‘mg‘itfaﬁ £14 :L we]? mg, (m? “5%. 5: E} 774'. 5'Mal§é5“§' fﬁﬂl'ff‘m? :‘Mj‘é‘iéng‘ﬁ nee-2d amt be equal m 11.: Z”, ﬁlm 2:; f a?» “PM ("mi i ,4 W [3'] m 2% W W W “3% FE ma Ufa Ejmve or (improve: if as?) m m: and a, 7% I) in Z,,_._ the}; I) 2 (2. 5;M‘%ww%~ i» m” ﬁlméééiﬁt" Math 310 ~ Section 3. Pal}: “201.0 Midterm 1 T. [10 113135.] Solve! tho equation 63' : a3 (mud 2?”). (WWW? W 3M SMLQ. kn} “3 ~:. 8‘ [10 pts‘] Assign a grade of A {camrrect}: C (partially correct}: or F (inﬂame) to the fhliowing “pmof.” Justify your assignnmm'. of grade. Claim. (For any :1, % [J in Z,“ that equation are: 1 has a 3011mm}: in 27,} m} {KN-7'imtlevm' a!) : U in 53;): than 5.1. 0 or b 0), “Proof”. 5113333050 £sz 2 U in Z”. If a 2 (.1. than we are done. 1:50, 74E {L than by ifl}-’p0tjflebii\$: 12111)]? fKiStH 21 U. such that (1sz m I. But if rm 2 '1. than ei‘ahm‘ (I. x u =2 'l., 0:: a, r; u m ——"i. In either case: ab if: z 0. \$9 5) must he 0 in Zip. “This tam"? wwﬂisw ﬂaws/Willi 9‘; 7521 MW alfmml? U mi? ‘i‘f'uét um W; E Bet-ma cum M E‘EW‘Mﬁ i“ 2F“ 5f ...
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