MATH 482: HOMEWORK 4
(1) Prove that
(xn/d 1)(d)
n (x) =
d|n
Hint: Take the log of both sides.
(2) Let p and q be distinct primes. Prove that
pq (x) =
p (xq )
q (xp )
=
q (x)
p (x)
Hint: Use problem (1).
(3) Find the winning rst move for Player 1 for 3 5 c
MATH 482: HOMEWORK 7
(1) Use Z11 to construct an Hadamard matrix of order 12.
(2) Give an example of a seven player chess tournament (where each player plays
every other player exactly once) with the following disastrous outcome: given
any two players a a
MATH 482: HOMEWORK 8
(1) Prove that R(3, 5) = 14.
Hint: The fact that R(3, 5) 14 follows from problem (5) on Homework 7.
To show that R(3, 5) > 13, consider the following graph: the vertices are the
elements of Z13 , and a is adjacent to b if a b is a non
MATH 482: HOMEWORK 9
(1) Find the eigenvalues of the adjacency matrix and Laplacian matrix of C5 .
Hint: There are only three distinct eigenvalues, and two of them have multiplicity two. Use the matrix tree theorem and the formula for the trace of a
matri
MATH 482: HOMEWORK 6
(1) Find all the elements a F9 such that ak = 1 for 1 k 7 and a8 = 1.
Hint: First use one of your answers from (5) on Homework 5 to construct F9 .
(2) You are on a game show. In front of you are 8 chessboards, numbered 1 to 8.
The hos
MATH 482: HOMEWORK 5
(1) Prove that xy + x + 1 = (x + 1)(y + 1) + (y + 1) + 1 in Z2 [x, y ]. What is the
interpretation of this equation in Boolean logic?
(2) On a certain island, each inhabitant is either a knight or a knave. Knights
always tell the trut
MATH 482: HOMEWORK 1
(1) Find the rightmost digit of 2123456789 . Prove that a positive integer is divisible
by 3 if and only if the sum of its digits is divisible by 3.
(2) Write down the multiplication table for Z10 . Which rows and columns have to
be r
MATH 482: HOMEWORK 2
(1) Prove that (n) is even if n 3.
(2) Find all integers n 1 which satisfy the equation 2(n) = n. Justify your
answer.
(3) Find the smallest positive integer n such that n (2a 3b ) = 1.
Note: n (x) means (. . . (x) where is applied n
MATH 482: HOMEWORK 3
(1) Let a be an integer such that a is not divisible by 7 and a 1 is not divisible by
7. Use Eulers theorem to prove that a5 + a4 + a3 + a2 + a + 1 is divisible by 7.
(2) Bob broadcasts a public key (n, e) = (1000, 29). Alice sends Bo
MATH 482: HOMEWORK 10
(1) Let n be a positive integer. Let S (n) be the number of self-conjugate partitions
of n. Let O(n) be the number of partitions of n which consist of distinct odd
numbers. Prove that S (n) = O(n).
Hint: Give a combinatorial proof vi