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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 MILMl, (J z .23 84133—34. (:23— 6/4, (33A 3 £22»
'3“ (23) 64x 32 (23)
E ~44: 1.233
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{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
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E \ (30) curl 6:53 3 5.4 or. awelimit 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‘fwﬂLi’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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 Fall '09

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