midsol - Math 1153 Spring Quarter Midterm Examination May 3...

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Unformatted text preview: Math 1153 Spring Quarter Midterm Examination May 3, 2007 I" I. ”I NAME: Amwa’b .. fl Please put you); name on the line provided above. You have 70 minutes in which to compiete this examination Attempt a1} of the questions AH questions cauy equal weight Note that you will not be awarded fukl credit on a question unless your answer is clearly, carefully and neatly stated. UP TO 5 BONUS POINTS (ADDED DIRECTLY TO YOUR SCORE ON THE EXAMINATION) WILL BE AWARDED FOR. NEATLY AND CARE» FULLY PRESENTED WORK! Scone -- NEATNESS BONUS (l)(i) State Euier’s Criterion concerning quadratic residues“ Lefi p be a Er flame, and. at am lidééaév wag (QM): .._... e: a (P) .‘ (D4294. L a, {Fwd/,1 {3 (ii) Show that 3 is a quadratic residue modulo 23, APPL‘ZTMS Eula“ Criétamw MIL-Ml, (J z .23 84133—34. (:23— 6/4, (33A 3 £22» '3“ (23) 64x 32 (23) E ~44: 1.233 a\ in) {Surcfl Ale 3 24 a @wcflrafic; «car/{fire wwwfi' =23. HI I” (iii) Show that 3 is a quadratic nomresidue modulo 31. APP‘T”? Eufer’s («MW «AWL ”rd 3!" «(1?de (E3 ~:—: 3”” ”"1 a 3”” (a) a 2(~4\5 (30 Ewaxfi an E -\ (30) curl 6:53 3 5.4 or. awe-limit Ww “Ce—HM W549. ‘31. (2)01) State the Quadratic Reciprocity Law f0: distinct odd primes p and q‘ <%\( ‘35) 1 (“‘{FmoWQ/é; P (b) Evaluate the Legendre symbol (71/73). m' % {$73) 2 ( ET) Na Wanawé't “mfi‘mfl‘é‘x (3)(a.) Evaluate the Jacobi symbol (3/185). w (ex ems» (b) Determine Whether 01" not the congruence 2:2 2 (mod 185) has any integer solutione (You must justify your answer) (D442 Wanna/ice 541‘; g (135) 42mm a. Solw’ér’m .; 2f 2‘3? 3 (-1-;qu :— \ gx'me ( E B: '— \ J H: ‘gVHWS M Me @fia’Tt/esgwge \QflS elm WLO gotufiims. (4) Suppose that n is an integel with n a £4 (mod 9)‘ Show that the equation n = a3 ~§« ()3 + (:3 has 110 integer solutions. (5)(a) Suppose that n, is a positive integer, with n = N 2m, whei'e m is squaiei‘reei State a necessary and sufficient condition that must be satisfied in order for n to be representable as a sum of two squares! (0/412 {0&223521 W ’er ~cQKlrfii-‘2bdéfl.&[€ LL14 6! Show,“ 0g 1’10? S‘fwfl-Li’éo ig mi 91413: g "W 42"“ "w” P“""“’* $5141“ '91} W gem/v 4Q+3, (b) Prove that no positive integer of the fowl 41% + 2 can be written as 3. ciifi'erenee of two squares. «Jig 0!. ('Ld (5mg “5'ij ”Lem, d2 :3— 0 of i (14:), Hence, 1g 1 seal 03 Core muff \Ldéian/(S, W H -f~ffl,0as\£fl, % l'vnyfiitiefl Wadi W )flélfiirwe M1348” a} W ZQWW‘ 4é+ol Cam. ice wwiHchx (M a. Jmhggflfgyue 03L W Sag/Midway ...
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