Number Theory
Spring 2013
Homework 4
Problem 1. Solve in Z the equation
35x 4
mod 4.
x4
x8
mod 18
mod 51
Problem 2. We consider the system
Prove that it has no solution.
Problem 3. We consider the sys
Number Theory
Spring 2013
Review 1
Problem 1. Let (G, ) and (H, ) be two groups. Consider the set
G H = cfw_(g, h), g G, h H .
We dene the operation on G H by
(g, h) (g , h ) = (g g , h h )
for g, g G
Introduction to Number Theory
Spring 2013
Homework 3
Problem 1. Let (G, ) be a group with identity element e such that
g G, we have g g = e.
Show that G is Abelian.
Problem 2. Let n N, n 1.
What is th
Introduction to Number Theory
Spring 2013
Homework 1
From the book:
1.2: Problems 8 and 16
1.3: Problem 24
1.4: Problems 6
1.5: Problems 6, 8 and 34
Problem 1. Explain why the following are not groups
Introduction to Number Theory
Spring 2013
Homework 2
Recall that a set G with an operation is a group denoted by (G, ) if
i. For all g, g G, we have g g G.
ii. The operation is associative: for all g,
Number Theory
Spring 2013
Homework 7
Problem 1. Consider an RSA system based on the prime numbers p = 13 and q = 7. We set
n = 13 7 = 91. We choose the public key (n, e) where e = 77.
1. Check that e
Introduction to Number Theory
Spring 2013
Homework 6
Problem 1. Let p be a prime number. Solve the following equation where x Z:
x2
mod 1
mod p.
(Think of using a2 b2 = (a b)(a + b).)
Problem 2. (1) L