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# Exam 01 - Math 2602 TESTl Name gem/UK Problems(P1(10 pts...

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Unformatted text preview: Math 2602 TESTl Name: gem/UK. Problems (P1)(10 pts) 1 (P2)(10 pts) I (P3)(10 pts) (P4)( 10 pts) [Total H Scores ’ TIME ALLOWED: 50 MIN SHOW YOUR REASONING TO RECEIVE CREDITS Problem#1.Decide which of the following Claims are true. Write True or False and Justify your answers. (a) 4n? + 12 : 0(2n3 _ 1) ‘1 E5 ‘. TWE , w _ L}{\1+lz , 1 .. 3" =5 Lln’L-HZ :0 2 }_‘ 3mg 2/3“!“ 933394 = 0 I Lm +\2 - 004» 3 > (n Yr)“, (b) 27;): 0(n2) NO \ VASE, J33“ Elizw m» n ”(ﬁnd :2; “3:09-33 Yl~>m VxL ‘ ‘5 2n3+0(“1\ (C) 2103< 3<)=o32)n WES‘ Tm: [\M 292030)} . : ltm ’13.: z.- 0 :_>> -l\03(n3) = 9(3VCL) W600 3%“ “am 3 1“ \ 3, . (d) 2103032): 3310332)) N° 2 2:2 Me ————2»-‘““ 2,2“ -333 W333)! (333)) “~59“ 310 2 m t (e) M + 8log(n9) + 4 = @(3n6 — 2) 7g§l 33333; (23 23‘233933 (3“) +1) .. :32. #0 :> 23% xhgh‘llw = @:%é~1l W505 ant a ‘2 Problem#2. (a) (5 points) Find the formula for _ j i 2 I n a”_Z(j+1)!*2!+3!+ +(n+1)!' i“ i”‘*‘ 2 > (RE: ‘ :” <7, “ 1-“ 53‘ {3‘ (9*\)\. 4“: 3 (3H) \ (\ . 3 _ K A- _._._. (b) ( 5 points) Prove your answer in part (a) by using induction. R 4. M4 _ ‘ 3,.) am: A— 1— :kwwl. LHS-(\‘H)\ : 21-2 ) , 1\ :1 2 LHS 2W5 \/ k ‘ \ 2.. : I- ’— was Jmn éz Lamp,“ 2 3W M 266)! GM» 5:! A+l 5 A , '1’ A M W. 6;; 477 k: m . 67m 7 " (WM, w“ 174 “’7‘ M-H * a U .L 32+: + 1‘; + —-——- 6:3 _, 2M + 3‘? 51‘“)! (mm ’3“ W 3 \ A Wt“ (Wm) (mt): _ wt mm ~ ._ ‘ .1. M (mat) 3 (“—616 §\ 5 ‘ (may! Problem#3. (a) (4 points) Calculate gcd(138,36) by using [Euler Algorithm. (’0 , \38 == \$936+" 30 3% / S'G ‘Q’O‘ , -' (3) (b) {3 points) Use part (a) to ﬁnd integers k and m such that 138k + 36m = gcd(138, 36). Cm ==> ‘4 r 34’ "‘ 30 :> (:96— (mr— 2130 .335 (43 ::> z 30 =9 ‘36 6:- 4.34438 30 \L2" ) mthf (C) {3 points) Use part (b) to ﬁnd the general solution of 23x"+ 6y = 1. WW) 1> 6: we» ~88 :5 «= Lt'é‘l' 23.90 WWI/{A ’ 0 Wk ) SmM1W‘3. 3: 304.33“ ‘3 Lf‘tQZV‘ 4 Problem#4. (a) (3 points) In a certain computer system, a ﬁle name must be a string of letters and oligits7 1 to 10 symbols long. The ﬁrst and last symbols must be numbers and the computer makes no distinction between uppercase and lowercase letters in any position. Thus 1amkA4 and 1AM ka4 are both the same legitimate name, whereas a1mkA4 is not legitimate. How many legitimate ﬁle names are there 7 Express your answer as simple as possible. /\ 9);“on 1P“) 1 1.0 garter A3 ' mmﬂ Wrrww‘ 3 WQ '. lOX’HXlO . H 9%:on 15% .1 \oxzéxZéxlo .. M0 w 3 Q a w 36x34 x 53M 1 W L} 3’ W5 \0+ \Ozt 36 \o"+ , 36" \o"+ -- a 363 lo =|0+l0 (b) ( 5 pomts) How many integers from 1 to 2007 are divisible by 3 or 5 or 7 ? Lilongwe? 6651f] LEM—ii: 40‘ L:m9—J= Q86 \Eﬁiaw Lam? :75 L391”; :10] :ttgrlmkjm {heat} 0/ 5 or a» is 66%er +234- tar—WMS +I‘i 2* mo ‘ (o) (2 points) How many integers from 1 to 2007 are not divisible by 3, 5 or 7 ? 1:} ”#55515 m ”A” Jim/MEL. L2, 3; § or 3‘ ——~ 0300?... [030 ...
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