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 and h, h H .
Prove that (G H, ) is a group.
Problem 2.
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 system
(S )
x 4 mod 18
x 10 mod 51
1. Prove that the rst l
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 the remainder of the Euclidean division of 1 + 2 + . + n
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:
(1) (N, +)
(2) (Z cfw_0, +)
(3) (Z cfw_0, )
(4) (Q, )
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, g , g G we have g (g g ) = (g g ) g .
iii. There is an
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 is prime to (n).
2. Compute the private key (n, d) for
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) Let (G, ) be a nite commutative group with unit e. Let X