MAS 4214/001 Elementary Number Theory, Fall 2013
CRN: 80553
Answers, Set 5
Qyestion 1. (2 pts) This problem is about solving ax b (mod m), where a = 2123 and m = 4632.
(a) Use the extended Euclidean A
MAS 4214/001 Elementary Number Theory, Fall 2013
CRN: 80553
Answers, Set 1
Question 1. (2 pts) The aim of this problem is to prove that n3 3n for all integers n 0.
(a) Show that n3 3n for n = 0, 1, 2,
MAS 4214/001 Elementary Number Theory, Fall 2014
Test 1
Date: 09/24/2014
Name:
Time allowed: 60 minutes
Show ALL steps. Maximum possible score 53 (out of 50)
(1 point)
Question 1. (10 points) True or
To prove that the converse is false, we need an example. Take I : 3 and a1 = 6,0,2 2 10413 = 15. Then
gcd(6,10,15) = gcd(21 15) := 1,
but gc(l(6, 10) x 2 1. [1
Remarks: In fact, the god of each pair o
Chapter 22
The Groups Um
Denition 22.1. Let m > 0. A residue class [a] Zm is called a unit if
there is another residue class [b] Zm such that [a][b] = [1]. In this case [a]
and [b] are said to be inve
(b) Prove that for m n, gcd(Gn, 4m) 2 1. (Hint. Use the result: gcd(bq +1',b) : gcd(r, 12)
Solution. (:1) We shall be using the equation $2 - 1 = (:c ~ 1)(a; + 1) repeatedly. For n 2 1, we have
n
Gn
cfw_.sws. .
Question 6. (1 pt) (True or false) Suppose that a, b are integers, not both zero. Let d = gcd(a,b). Assume
that there are integers s and t so that
18 = as + bt.
Prove or disprove the fol
Chapter 19
Residue Classes
Denition 19.1. Let m > 0 be given. For each integer a we dene
(1)
[a] = cfw_x : x a
(mod m).
In other words, [a] is the set of all integers that are congruent to a modulo
m.
Question 3. (1 pt) (Exercise 7.1.)
(a) Prove that d I a implies that (l I a.
(b) Prove that d I a iff d I a. '5
(c) Prove that d I (1 iff d I IaI.
190171177071. (3.) Assume that d I (L. Then there is
giving that
ac=~3w+5k, y=2w3k, kEZ.
Since in = 56 + 15h7 the general solution is
9;:16845h-i5k, y=112+30h3k, 2:8211, h,keZ.
Please note that the solutions can appear in other formats.
(1) (1.5 pts) We
MAS 4214/001 Elementary Number Theory, Fall 2015 CRN: 80518
Answers, Set 4 (There are 11 points and 10 points equal 100%)
Question 1. (1 pt) Consider my solution to Exercise 13.4 in Assignment 3 (to
Chapter 23
Two Theorems of Euler and
Fermat
Fermats Big Theorem or, as it is also called, Fermats Last Theorem states
that xn + y n = z n has no solutions in positive integers x, y, z when n > 2.
This
(3) True, because of the cancellation property with gcd(12, 15) = 3. The converse is also true.
(h) True, since a; E 73 (mod 75) is the same as saying that a; and 73 have the same remainder when divid
A ~4va._
Wwww*v_v7w 7
By FLT, we have
(11 E 1 (mod 5), (11;2 E 1 (mod 13) am E 1 (mod 17).
Since 771 -~ 1 = 1104 is divisible by 4, 1:2, and 16, we have for some integers ([1,(]2,cfw_13 that
(rm1 = (
cfw_(7) Find all solfiuveises modulo m. = 1417 a (3) (49).
Solution. (:1) cfw_:>" part.) Suppose that, x is a selfinverse, moudlo 1), Then 1'2 E 1 (mod ,0"), giving that
p l cfw_:17 1)(~1: + 1).
By Eu
MAS4214: Practice problems
November 1, 2016
Question 1 We want to find all self-inverses modulo 3720151123 .
Let p be a prime. Prove that if p | (x 1) and p | (x + 1) then p = 2.
Prove that if 37 |
Chapter 27
The RSA Scheme
In this chapter we discuss the basis of the so-called RSA scheme. This is
the most important example of a public key cryptographic scheme. The RSA
scheme is due to R. Rivest,
Chapter 18
More Properties of
Congruences
Theorem 18.1. Let m 2. If a and m are relatively prime, there exists a
unique integer a such that aa 1 (mod m) and 0 < a < m.
We call a the inverse of a modul
MAS 4214/001 Elementary Number Theory, Fall 2015 CRN; 80518
21
Answers, Set 2 (There are ,Llfioiuts and 10 points equal 100%)
Question 1' (1 pt) (Exercise 5.4) (a) Prove that if a | b or a I c, then
Chapter 15
Congruences
Denition 15.1. Let m 0. We write a b (mod m) if m | a b, and
we say that a is congruent to b modulo m. Here m is said to be the modulus
of the congruence. The notation a b (mod
Chapter 26
Computation of aN mod m
Lets rst consider the question: What is the smallest number of multiplications required to compute aN where N is any positive integer?
Suppose we want to calculate 2
MAS 4214/001 Elementary Number Theory, Fall 2014
TEST 3
Name:
Date: 12/01/2014
Time allowed: 60 minutes
Show ALL steps. Fifty points equal 100%.
Assume in this test that all small Roman letters repres
MAS 4214/001 Elementary Number Theory, Fall 2014
CRN: 80544
Answers, Set 1
Question 1. (2 pts) The aim of this problem is to compare the values of 3n and n3 + 1 for dierent values of
n.
(a) By constru
MAS 4214/001 Elementary Number Theory, Fall 2014
CRN: 80544
Assignment 1
Question 1. (2 pts) The aim of this problem is to compare the values of 3n and n3 + 1 for dierent values of
n.
(a) By construct
MAS 4214/001 Elementary Number Theory, Fall 2014
TEST 2
Name:
Date: 10/23/2013
Time allowed: 60 minutes
Show ALL steps. Fifty points equal 100%.
Assume in this test that all small letters represent in
MAS 4214/001 Elementary Number Theory, Fall 2014
CRN: 80544
Assignment 2
Question 1. (1 pt) (Exercise 5.3) Prove that if a Z then the only positive divisor of both a and a + 1 is 1.
Question 2. (1 pt)
MAS 4214/001 Elementary Number Theory, Fall 2014
CRN: 80544
Assignment 4
Question 1. (0+2 pts) (a) Find all integer solutions to 3x 9y + 15z = 7.
(b) Find all integer solutions to the system of equati
MAS 4214/001 Elementary Number Theory, Fall 2014
CRN: 80544
Assignment 5
Qyestion 1. (2 pts) This problem is about solving ax b (mod m), where a = 2123 and m = 4632.
(a) Use the extended Euclidean Alg
MAS 4214/001 Elementary Number Theory, Fall 2014
CRN: 80544
Answers, Set 6
Qyestion 1. (2+1 pts) Similar to (but a little harder than) Exercise 23.3.
(a) Use the CRT to solve for x, where a, b, c are