graded HW 2

graded HW 2 - Math 300, Fall 2011 Homework 2 “‘1 Due...

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Unformatted text preview: Math 300, Fall 2011 Homework 2 “‘1 Due Monday 26 Sept, 2011 RP» O 1) Write thicmitrapositive‘of the implication = 2 = m = 2 07' m = —2. Mad? . 2) Let n E Z.\Pfove‘by/(:ontradiction that there is no a: E Z such that n < x < n+1. /f 3) Find a composite number n > 5 such that n divides (n ~ 1)!. ‘45 4) Let n E N. Prove that if n divides (n — 1)! then n is composite. Jw 5) Let n be a positive integer that has 6 and 8 as factors. What other factors must. n have? v6) Prove that if n is composite, n = kl, with 1 < k < n, 1 < l < n, then It and l both divide (n — 1)!. \A 7) Find a prime number p and an integer b < p such that p divides b(p_1) — 1. A 8) Let p be prime and let 7; be a natural number. Prove that p cannot divide both n and n + 1. v9) Let n be a natural number with prime decomposition n = pilp? H.102". Prove that n = m2 for some natural number m if and only if 31, 32, . . . , 5k are all even. and n > 1 then a: = 2 and n is prime. (10) It is known that if 2” - 1 is prime, then n is prime. Prove that if at” — 1 is prime % ., 1 flipgfi MM its pfi‘ come 11% UP teem comm”) I m \csé M“ ‘m‘ J! N4 =‘7 X=Z ov m: 4; @3933 {gig/amt?) 10$: 9%.. 031% mm 5 , Law p: mm (51% m 1W2. Wok "Iv—1.3m M «=2 =‘1 m. mar—.2- 1 W2 and”! 731 fi 7 \’)K\=2. é 4m 15‘3m15516 You s‘nowk 3% 0L WW’M . Lat n ,xmm e1: mm m<"l\“{\*\ .msumamgmwwi Aways .. x3. L) $0, NS Wen . ’k-Y\*\,L22L..LOO.W \S mfivmfiéwfi < WERSUMW’C a? MAWQN “mm \‘s mm .. 9mm \\ \s \m§\w_ w I xx kavm W make“; mmm§mmmm mwm may) \8 own ‘3 A mypsfla mwmr \n > 5 won “M292: 0 woos vaO‘. “Tm msvcmmfimmwfiwmw 51s :9 +30 \m “m- ,(vx-x)! = . {nu-0|:— (53‘.=.S_°1+-3~2r 7:039? \Zbfi mule Hfmg‘ L0 is. .a ‘31ng Li _. propa§fi\m= m m: N flmfina¥ xx. \(\ Wm (WM, ,,,.qu VHS. mwa- 7%me meW-,fiMm MMW\ 9:3 CNN $43ch 6N“, \I\ hm ‘xmcsxm A. V? a wmw \5. m Wm X}. wwr m gawk. mm. ,‘x& .3} \mm ‘ \(\ mm xx mmme mwae a, ytmmmd \n <waka am. um)“ \im W’J’WWPN) m cm M3 MW! 5. m w m a 935m: mmer kw m3. :9 am “6 as Edam mm: 6W Seam: g M new!) .\¥ um } 1&2 mmmx w— Lmfi— mammm \A wfiw dammmm 1M am am- \X mm. 3 méfiswn WW w- .i .1: www- Lflfltt‘im.mm\n vhm mmm m w Mm. WRMQS m mam wmm 1.3; .4 - \_,\’z.r, 9J1 - quoWMNWQ’mfl “gmkmvm mm “Km M “.X‘“ v\ gm 1 mm. aim Lw-m. A H. isMthm» Ln—Q‘, = Lnflmfllwa): I, QOYHWLS mm wax-e = i 4 W limqu Q. N 3% VFW.- m SM 3‘va ’XV m; M. magma J<K_<V1 *fi myriad“ M Him 9 “rd-114w 8111.1 mamfimw SmaerWug mm: (VI-1.3Ln-mYn-131wq’(n-\)l=.(KXxYn-h).w [G N- ' m» Ymfi film-0mm MUM)! Mn n is, “7’0 -1 Trim amnst MW x2 ammx may bf? mme m b -1. mm, MW 5 \nmqu K4 c q,5“«\ '- H‘flr 2519': = 2.55.: 5 \156, y W WWW m aw WI Magma W “1&9ng it gum WWWG // ;% Lyopmmm '- Lat mwm «m mm m .a Meet/WAVE my; mm -dimmwéndmh f % - 1831933359. 1>ln=‘-“L.€. $00k, plfi4|= MEN. we 14me mmLQemfim gm mam exam/2 am; MM 862% and axflnatt @65va mm and 04¢; AH pvm, “mm; £19293er m cm (aim “MM 4%" w- mm 5h m mm wmw W\ ’6; am 0m \82 S.,$L-..3k6fid‘\\fi£fi‘ ml? n=m"= mm. m max we, a Wm. WM“ chfifitwti .Tmm - a 4‘ a, .I: h log {z 1); ’15:; 23;» WW9. damvgosmm 0% w—M 4%. 0&1 momma "w ‘- amt. “Aw. . Rm , qf‘qf‘i- ,oyf“: Y? ??,gg?f‘..,em mmm Mmgggsfim Q? m amvt . mw. \me f‘mWa/x Jmmm Q? emwnm, ,. am am, WW3 . mm ‘\S,\x\\,o\u<, $1) cut-pg and Zh=81,wm slwwgk . rag dmm m3 1, gm ks ‘ K) Qmfimz \fi \8 \fwix Z“~\ \3 QfiW‘W ms mmfim Mk \S; wqjsfimw amwa “M 6m WEQAW‘. M \s M)me &\\ Wm aw; mi mafia: 1 .mom.m W: \M. “m. m mmmmse mm m 1“-\=1 Mm omw wummmmsmx 4 33¢st n>\_.w\ an: $35me H Q‘WeW 3w» \‘< ww 36,qu Odd / "RM:— _,Qe_?%.0_mflf=4y\.€ mass? m mo? 3mm mtgan ‘D ,- VAR-32y: @Mnaxe we“ em‘ odd em m \mfiwwm mxgml X 30., «mm; M to a“ :m Wm MR mum“. mm wmw z» m = 6:; ¥cv m 9mm 3m“ mwrmiety gm mm swam ,x «w V em mm “mm %\‘(\S\\_&§‘V $3M max “mm 1am} PT’Rvax can 64101»: mow ...
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graded HW 2 - Math 300, Fall 2011 Homework 2 “‘1 Due...

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