200953621 - University College Dublin An Coiéiste(Jimmie...

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Unformatted text preview: University College Dublin An Coiéiste (Jimmie. Baits Atha Cliath SEMESTER 1 EXAMINATION 2009/2010 MST 10040 Combinatorics and Number Theory Professor P. J. Rippon Dr. Micheal o Searcoid Dr. T. Unger Dipl. Math. C. Rofiing* Time Allowed: 2 hours Instructions for Candidates Full marks will be awarded for complete answers to all four questions. Instructions for Invigilators Non—programmable calculators may be used during this examination. ©UCD 2009/ 10/ Modular 1 of 2 $362! n 1. (a) Expand the summation Z 2i. i=1 (b) Use the method of mathematical induction to prove that V: zi = 2”+1 ~ 2 i=1 ' for all n E N. (c) Find the first seven terms of the sequence a1, a2, . . . ,an where 0.1 = 3 and an = 7 ' an_1. (d) Proof by mathematical induction that an = 3 - 7”“1 for all n E N. 2. (a) Let 71,7" 6 N with r S n. Express in factorial notation. (b) Showthat n _ n—r n r+1 _ r+1 7“ I (c) How many license plates consist of from zero to three letters followed by from zero to four digits where the all blank plate is not allowed? Suppose 85 letter combinations are not allowed because of their potential for giving offense, how many different plates are possible? (d) In a simple language there are the usual 26 letters, but all words have exactly 4 letters. Any arrangement of the letters, including repetition, is allowed. How many words are there? How many words would that language have if there were also 1, 2 and 3 letter words allowed? 3. (a) State the binomial theorem for (a: + y)”, n E N. (b) Use the binomial theorem to find the coefficient of a6b6 in the expansion of (a2 + b3)5. (0) Which decimal number has the octal representation (123)3? What is the hexadecimal (base 16) representation of (2009)10? 4. (a) Use Fermat’s Little Theorem to find the smallest positive integer n sat— isfying 290 E n mod 47. (b) Find all mutually incongruent solutions of the linear congruence 142 t as E 16 mod 410. (c) Use the Chinese remainder theorem to find the smallest positive integer satisfying the following system of linear congruences: 4mod5 6mod7 2mod6 ll! ©UCD 2009/lO/Modular 2 of 2 ...
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