Math 311M Homework 5
Fall 2011
Due: Thursday, September 29, 1pm
(in my mailbox in 109 McAllister)
1. In the decimal (base 10) system any positive integer a is written uniquely in the form
a = (a)10 = an 10n + an1 10n1 + + a1 101 + a0 ,
where a0 , a1 , . .
Math 311M Homework 2
Fall 2011
Due: Friday, September 9
1. Prove each of the following statements using induction.
(a) Bernoullis Inequality which states: If x 1 then (1 + x)n 1 + nx for all
natural numbers n.
(b) For all natural numbers n 1, 4n+1 + 52n1
Math 311M Homework 8
Fall 2011
Due: Friday, October 28
1. Prove that a simple lattice polygon with at least 4 vertices always has an interior
diagonal, i.e. a diagonal that is entirely inside the polygon. (A diagonal is any line
connecting two non-adjacen
Math 311M Homework 9
Fall 2011
Due: Thursday, November 3, 1pm
(in my mailbox in 109 McAllister)
1. The order of a permutation is its order in the group S (n), i.e. the smallest positive
integer k such that k is the identity permutation.
Let be a cycle in
Math 311M Homework 10
Fall 2011
Due: Friday, November 18
1. Prove that, up to isomorphism, there are only two dierent groups of order 4.
2. Prove that the set of all even permutations of the set cfw_1, 2, . . . , n forms a subgroup
of S (n). This subgroup
Math 311M Homework 11
Fall 2011
Due: Friday, December 2
Please explain all of your solutions, including the questions about counting, using complete
sentences.
1. Recall that Z is the set of invertible congruence classes modulo n.
n
(a) Prove that Z is cy
Math 311M Homework 7
Fall 2011
Due: Friday, October 21
1. Compute the canonical representative of 20112002 in Zm for m = 17 and m = 100. Find
the order of this element modulo m in each case.
2. Does the equation x2 + 3x 3 0 have a solution modulo 5? Does
Math 311M Homework 6
Fall 2011
Due: Wednesday, October 12, in class
1. Let p be a prime. Prove that each prime divisor of 2p 1, is greater than p. (Notice that
this provides a dierent proof of the theorem that the number of primes is innite!)
2. Prove tha
Math 311M Homework 1
Fall 2011
Due: Friday, September 2
1. Let a and b be two positive integers, and gcd(a, b) be their greatest common divisor
obtained by the Euclidean algorithm. Prove that any common divisor of a and b divides
gcd(a, b).
2. (a) Given p
Math 311M Homework 3
Fall 2011
Due: Friday, September 16
1. Guess and prove a formula for 12 22 + 32 42 + + (1)n1 n2 .
2. On a circular road there are n gas stations. The total amount of gas is sucient to
make the whole circle, but an individual gas stati
Math 311M Homework 4
Fall 2011
Due: Friday, September 23
1. Show that there are innitely many primes of the form 4k + 3 (with k N).
2. Prove
that
any
subset
of
55
numbers
chosen
from
the
set
cfw_1, 2, 3, . . . , 100 must contain two numbers diering by 10
Math 311M Homework 12
Fall 2011
Due: Friday, December 9
Please explain all of your solutions using complete sentences.
1. An elevator starts at the basement with 8 people (not including the elevator operator),
and discharges them all by the time it reache