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Unformatted text preview: Name (Print): MW
UW Student ID Number: 4 \ University of Waterloo
Second Midterm Test Math 135 (Algebra for Honours Mathematics) Instructor: RD. Willard Section: 001 Date: Monday, March 6, 2006 Time: 7:15 p.m. to 8:30 p.m.
Term: 1061 Duration of test: 75 minutes
Number of pages: 7 Test type: closed book (including cover page) Additional Material Allowed: Math Faculty approved calculator (Sharp EL—531W with pink tie
sticker) Instructions 1. Write your name and ID number at the top
of this page. 2. Answer the questions in the spaces provided,
FOR EXAMINERS’ USE ONLY using the backs of pages for overﬂow or
rough work 3. Show all your work required to obtain your
answers. 4. Only Math Faculty approved calculators
(Sharp EL—531W with pink tie sticker) are
allowed. Math 135 Second Midterm Test, Winter 2006 Page 2 of 7 Name: [5] 1. (a) Find the inverse of [23] in Z168. Express your answer in the form [a] where 0 S a S 167. / """ [VIZ]: U] (ml (633 Lg) 23% t (UL/101 l”) 13553 (131%?!) H ‘Salm {mm o x 5 Z
'3 22 2 3
10 33’ l 3 0 Z (BX/XML (650(10):
“WWW? eel—MM Mlht‘m T? Xi: ”13+ IL‘M néz _m _.. W”..— [@513 ml C ’/13(mw#lL€) °V‘ ‘zszl amwlll‘) [i330' l3] (l0) Given that a: = 8 is one solution to the congruence 30:12 E 42 (mod 99), ﬁnd the complete solution. Express your answer in the form a: E . . . (mod 99). I WWW”! 59/“ 710 ﬂ 9M4” army/Mm“ WéEC/Ma/m) p :é *’ 7% + aim mtg/r % :7Z/[h/I Mr) ,
wawgw, Iauﬂ/au Jwt />; j (2”gg”ww ; 9 I V) / «0f 30 g, 75: 34 jZ—n , 9+ 33 f. q; Math 135 Second Midterm Test, Winter 2006 Page 3 of 7 Name: [7] . 2. Solve the following system of congruences: a: E 1 (mod 39)
x E 12 (mod 17). xxx/«ﬁg» We? « /4 37:1 a /Z (64/191 /%)g 395 2 l/ (MM/y/ LUMP/€146 5.9% ¢ A 9 W >4: t + 39(}+l?n) : 1+ 2257+ 55%,; ,,,,,, 2613““ bE‘Jhm i
ﬂaw/m 1m) _ 7 E CUM/L 2 ZQ’Ll—f 651514 E I (ml 3'9)
l+ (mm. 2 (mm W) V Q‘szx  ~ @ Math 105 Second Mluncnn Test, Winter 2006 fags q of T Name: [2] 3. (a) Carefully complete the following deﬁnition: a E b (mod m) if . . . _
{ﬁve/l A {\b ‘9 Z aml ‘Vm l? a “Niel @5344, W ”infeﬂevj
‘2’ mﬂgpb )/ [5] (b) Carefully prove Proposition 3.11(iii): If a E b (mod m) and b E 0 (mod m), then a E 0
(mod m). &E 5 (MM “7) imp/{PI I’M/ﬂyé
6: (zonal m I‘M/far m/é—c V”, LL, ¥’4e ﬂ/rﬂ/mﬁibw If 4: égmo/ﬁ) ﬂan W? WW? ”3 ”WV [mg/hm” a/ [4] (0) Suppose a2 E b2 (mod m) and gcd(m, a+b) = 1. Prove that a E b (mod m).
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m/ [51%) (W4)
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116557)» 0/ mdu/ﬁ*””‘ Wymew [:7 Math 135 Second Midterm Test, Winter 2006 Page 5 of 7 Name: 4. In each part of this problem, full marks will be given if the correct answer is written in the box.
If your answer is incorrect, your work will be assessed for part marks. [3] (a) What is the inverse of [5] in Z13? [3] (b) What is the remainder when 3100 is divided by 7? \l \/ i3
ANSWER: ’ ‘l /A
3’": 3" (36(6) ; 3V0) [Mm/1.?) Do
’5’ 2 3" [mm ?)
(ﬂ: 9/ E Ll (Wise
. M E
[4] (c)\What is the remainder when f xx
. M ﬂ, 10123456789876543210 \
is divided by 55? ’ l 26?, ’ (ll) (”0’4 3C) ‘3
5 /’ (xvi/D C7
20 4 ‘>
ANSWER: g
4 0 W7 m <2) W WWW,“
4; C/WM M) 0:52 ._ Maw 1/) 4
(0 n/ a’ffclm‘oz! 57 0;:0 2:9(Wvlu)
721.; v. 0 7C 2 0 (014171 f) ,
‘2 l ?
:' ”5‘ N v\ ’
as 79’ C ( i w ’3’” 7x; €44 aim—US?
‘ (RH ”03:? M‘ m _’ inf/“r is?“ ln
5 d 5)+ m) :7 W: "9*“? m 9 W“ ~\ Math 135 Second Midterm Test, Winner 2006 Fage 0 Of T Nanié: “\ [10] 5. Solve x2 E 1 (mod 63). X ﬁrm LAW \ Gama {5H \
f 1 195 (MT. . 5’ WV 3 25"” / m ,[55] <::> \ Zﬂghffﬁ, gum/(4;) 7C 2 Z ) 0”) 4W! 9) g: _ VVVVV ‘f,’
3 ﬂ} 5. / (MM 373 : (Meek ‘9) {0/V6 @ x21 (mod 9) I @ 1:: WWW! g)
263 (Km/71 3) / 24: Mm” 7)
7" "+91% 3 (#133104401?) I] 5} ”WWW: 7a: a R
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M: /+ 9(%\ 2 / + 951/. ,1; / (Mod 8) j , _ _____ _, r/ @ 70? //[W05{ 9) j;&: /+ 93
706 1» (WM 5}) ;/+ (/jEéfmwy) 3.33% 6, q . >6 (“ward 7’)
Kg +21% if) (4:75, 7c: 8495
\nwww... «/ __ /? L; I 3 q / ?.\ , 0 ‘3», 3+ 3 : (7 vax , 4:433 —7 (mm! 2) 223,...” “if“ 7h) 3 WWW” 6E} 59%? a/ 724 097/ r» yf a»: Z 5 {y (33m: M y : 4H 43 ,m// if”: 8* q <~ *9 +3M 3312 W633»,
% E — D 33(qu 14;) 7(3 447mm 4 ?)‘ Math 135 Second Midterm Test, Winter 2006 Page 7 of 7 Name: [Blank page. This page may be removed for scratch work] ...
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