200851317 - fiamsity allege-Dahlia 1a fiaiie kmzct’iath...

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Unformatted text preview: fiamsity allege-Dahlia: 1a.: fiaiie kmzct’iath SEMESTER 1 EXAMINATION 2008/2009 MST 10040 Combinatorics and Number Theory Professor P. J. Rippon Professor S. Dineen Dr. T. Unger Dipl. Math, C. Roessing‘“ 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 2008/ Q/Modular 1 of 2 5131:!“ _._1_. 2' - (7L + 1)' (b) Use the method of mathematical induction to prove that 1. (a) Expand the summation 2 i=1 Lemmy; i=1i-(i+1)_ n+1 for all n E N. (c) Find the first seven terms of the sequence 04, a2, . . . , an where a1 = 2 and an“ = a7, + 2n. (d) Proof by mathematical induction that an = n2 — n + 2 for all n E N. 2. (a) Let n, m E N with m S n. Express in factorial notation. (b) Show that (1:) =(";1)+(::::>- (c) In how many different ways can you arrange the letters of the word FLAB— BERGASTED? In how many of these arrangements are the letters F and L side by side, as FL or LF? ((1) 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 last two digits of 1996 = (20 — D96. (c) Which decimal number has the binary representation (1111101000)2? What is the hexadecimal (base 16) representation of (2008)“)? 4. (a) Use Fermat’s Little Theorem to show that 11 divides 95 — 45. (b) Find all mutually incongruent solutions of the linear congruence 120 - as E 63 mod 321. (c) Use the Chinese remainder theorem to find the smallest positive integer that satisfies the following system of linear congruences: a: E 3 mod 9 a: E 2 mod 11 a: E 1 mod 13 ©UCD 2008/9/ Modular 2 of 2 ...
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