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Math 315; Homework # 5
Due March 4, 2015
1. (Exercise 18.1) Decode the following message, which was sent using the modulus m = 7081 and the exponent k = 1789. (Note that you will rst need to factor
m.
3 Test 1
3.1 The Questions
There are two parts to the tat. Part A has 10 quations and is mul-
tiple choice. Circle the letter corresponding to your answer. Part B
has 6 quations For PART B, write out
3
Divisibility
3.1 The definition of divisibility
If m and n are two integers, we say that m divides n (and write m|n)
if n = mk for some integer k. We have already used the terminology
and the notati
University of Toronto
Faculty of Arts and Science
DECEMBER 2016 EXAMINATIONS
MAT315H1F Exam
December 15, 2016
Duration: 3 hours. No aids permitted.
Last name . . . . . . . . . . . . . . . . . . . . .
4
Primes and Composites
4.1 What is a prime?
A prime p is an integer > 1 whose only positive divisors are 1 and
itself p. A natural number that is not prime is called composite.
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Congruences
5.1 Definition of congruence
Given a positive integer n, we say that two integers a and b are
congruent modulo n, written
a b (mod n)
if n divides a b. Thus, for example, 1 is congruent
MATH 315; HOMEWORK # 1
Due Jan 19, 2015
1. (page 11, Exercise 1.3) The consecutive odd numbers 3,5, and 7 are all primes.
Are there innitely many such prime triplets? That is, are there innitely many
Assignment #1 STA457H1S/2202H1S
due Wednesday February 3, 2016
Instructions: Students in STA457S do problems 1 through 3; those in STA2202S do all 4 problems.
1. Daily Japanese yen/US dollar exchange
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MATH 315; HOMEWORK # 1
Due Jan 22, 2016
1. (similar to Exercise 3.3, page 24) Find a formula for all of the points on the
hyperbola x2 2y 2 = 1 whose coordinates are rational numbers.
2. (similar to E
2
Some Interesting Sequences
2.1 Fibonacci Numbers
Consider the sequence a1 = 1, a2 = 1, a3 = 2, a4 = 3, where the
general term is given by the recursion
an = an1 + an2
for n 3. This is clearly a subs
MAT 315 PROBLEMS ON CONGRUENCES
(1) Prove that 7x10 + 2 = y 3 has no solutions in x, y Z.
(2) Compute the greatest common divisor of
(a) 5123 and 3215.
(b) 1235 and 2153.
(c) 2351 and 1532.
(3) Find i
MAT 315H1S
TUTORIAL 2: SECTION B
(1) Use the Euclidean algorithm to compute the greatest common divisor of the
following pairs of integers:
(a) 235 and 791
(b) 532 and 197
(c) 27128 and 31415
(2) Veri
MATH 315; HOMEWORK # 3
Due Feb. 4, 2015
1. (Exercise 11.2) (a) If m 3, explain why (m) is always even.
(b) (m) is usually divisible by 4. Describe all of the ms for which (m) is
not divisible by 4.
2.
MATH 315; HOMEWORK # 4
Due Feb. 25, 2015
n
1. (Exercise 14.2) Let Fn = 22 + 1, so, for example, F1 = 5, F2 = 17, F3 = 257
and F4 = 65537. Fermat thought that all of Fn s might be prime, but Euler show
UNIVERSITY OF TORONTO
DEPARTMENT OF MATHEMATICS
MAT315H1S
Introduction to Number Theory
Winter Semester 2015
Instructors:
Lectures:
Textbook:
Dr. Henry Kim, BA 6250, 416-978-4443
MWF9
A Friendly Intro