Applied Combinatorics With Problem Solving by Bradley Jackson, Dmitri Thoro

By Bradley Jackson, Dmitri Thoro

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Y>. Congruence Modulo n An important relation on the set of integers can be defined using division. Two integers x and y are said to be congruent modulo n, written x y (mod n), if they both have the same remainder when divided by n. Equivaleiitly x and y are congruent modulo n if x y is divisible by n. Congruences are a powerful tool for attacking many problems in number theory. It is easy to see that congruence modulo n satisfies the three basic properties of an equivalence relation: [Reflexive Property] 1.

Show that any positive integer i 1 can be represented uniquely as a sum of the form a11! + a22! + 1)! , where a1 is an integer + . 8 1. , P. Four holes are punched in the upper right-hand corner of each card (as shown in Fig. 8). Sixteen 3 x 5 index cards are labeled A, B, C,. 9 If the cards are in random order, we would like to return them to alphabetical order using the following procedure. Prepare the cards by slitting some holes as shown in Fig. 9. ) The cards are stacked in a deck, which is held vertically.

COMBINATORIAL PROBLEM SOLVING I Let a(n) be the number of representations of the positive integer n as a sum of l's and 2's, taking order into account. Let b(n) be the number of representations of n as a sum of integers greater than 1, again taking order into account (and counting the sum n = n). Thus a(4) = 5 and b(6) = 5. a) Show that a(n) = b(n + 2) for every positive integer n by describing a oneto-one correspondence. b) Showthata(1) = 1,a(2) = 2,andforn > 2,thata(n) = a(n 1) + a(n —2). 17.

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