Math350
Homework Set 2
1. Prove that if (a, c) = 1 and (b, c) = 1, then (ab, c) = 1.
2. Show that if (a, b) = 1, then (a b, a + b) = 1 or 2.
3. Show that if ad bc = 1, then (a + b, c + d) = 1.
4. Show that no integer of the form n3 + 1 is a prime if n 2.
Math350
L11. Chinese Remainder Theorem (4.3)
Feb. 6, 2015
Goals: To solve systems of linear congruences in one variables by using the Chinese Remainder
Theorem.
The Chinese Remainder Theorem. Let m1 , m2 , . . . , mr be pairwise relatively prime positive
Math350
L12. System of Linear Congruences (4.5)
Feb. 9, 2015
Goals: To solve systems of linear congruences in many variables.
Recall. Let m1 , m2 , m3 > 0. For a1 , a2 , a3 Z, if m1 , m2 , m3 are relatively prime, then we can
x a1 (modm1 )
x a2 (modm2 )
Math 350
L6. Factorization Methods and Fermat Numbers (3.6)
and Linear Diophantine Equations (3.7)
Jan. 28, 2015
Goals. Introduce the Fermats factorization method, Fermat numbers, and the concept of a
diophantine equation.
Fermat factorization of odd inte
Math350
L9. Congruences and Modular Arithmetic (4.1)
Feb. 2, 2015
Congruences
Denition 0.1 Let m be a positive integer. If a and b are integers, we say that a is congruent to b
modulo m if m|(a b), which is written by a b (mod m).
If m does not divide (a
Math350
L5. The Fundamental Theorem of Arithmetic (3.5)
Jan. 26, 2015
Goals. To understand the fundamental theorem of arithmetic and the multiplicative structure of Z.
The Fundamental Theorem of Arithmetic. Every positive integer greater than 1 can be wri
Math350
L10. Linear Congruences (4.2)
Feb 4, 2015
and Chinese Remainder Theorem (4.3)
Goals: To nd solutions to linear congruences, and to solve systems of linear congruences in one
variables by using the Chinese Remainder Theorem.
Denition. For a, b Z, a
Math350
L0. Preliminaries (Chap. 1)
January 12, 2015
Review. Well-ordering property, Algebraic numbers, Mathematical induction
Well-ordering principle.: Every non-empty set of positive integers has a least element. That is, For
every set S Z+ , there exis
Math350
L1. Divisibility (1.5)
January 14, 2015
Goals. To learn the concept of divisibility of one integer by another, and to study the multiplicative
structure of the integers.
Denition. (Divisibility) We say that a divides b or a is a divisor of b, or b
Math350
L2. Prime Numbers (3.1) and Distribution of Primes (3.2)
Jan. 16, 2015
Questions. What are the primes and composites? How to determine if a given integer n is prime or
composite?
Denition. (Prime number) A prime is a positive integer greater than
Math350
L3. Greatest Common Divisors (3.3)
Jan 21, 2015
Goals. To learn more about the greatest common divisors of tuples of integers.
Theorem. For any positive integer n, there are at least n consecutive composites.
Proof. Consider the n consecutive posi
Math350
L4. The Euclidean Algorithm (3.4)
Jan. 23, 2015
Goals. To compute gcds and express them as a linear combination of the integers by using Euclidean
algorithm.
Theorem. Given a, b Z, the gcd (a, b) is unique.
Proof. Because we dene gcd as the greate
Math350
Homework Set 2 (Sample Solution)
Due: Feb. 2, 2015
1. Prove that if (a, c) = 1 and (b, c) = 1, then (ab, c) = 1.
Sol. (Direct proof.) We have ax + cy = 1 and bs + ct = 1. Thus,
ax bs = (1 cy)(1 ct) = 1 c(y + t + cyt),
or equivalently,
ab(xs) + c(y
Math350
Homework 3 (Sample Solution)
Feb. 9, 2015
1. Prove that the product of two integers of the form 6n + 1 or 6n + 3 is also the form of 6n + 1 or
6n + 3.
Sol.
(6m + 1)(6n + 1) = 6(6mn + m + n) + 1
(6m + 1)(6n + 3) = 6(6mn + 3m + n) + 3
(6m + 3)(6n +
Math350
HW 1 (Sample solution)
Due: January 26, 2015
1. Let a, b, c Z. Prove each of the following.
(a) If a|b, then ac|bc.
Ans. a|b b = ax for some x Z; and so, bc = acx ac|bc.
(b) If ac|bc and c = 0, then a|b.
Ans. ac|bc bc = acx b = ax a|b.
(c) If a|b
Math350
Homework Set 3
Due: Feb. 9, 2015
The problems #1 through #4 will be graded for this homework set. The rest of the problems will be
optional.
1. Prove that the product of two integers of the form 6n + 1 or 6n + 3 is also the form of 6n + 1 or
6n +
Math350
Homework Set 1
1. Let a, b, c Z. Prove each of the following.
(a) If a|b, then ac|bc.
(b) If ac|bc and c = 0, then a|b.
(c) If a|b and b|a, then a = b.
2. Prove that if n is odd, then n2 1 is divisible by 8.
3. Prove that if 2p 1 is prime, then p
Math350
L13. Exam Review and Problem Discussion
Feb. 11, 2015
Goals. Review some problems from HW 3, and solve a few other problems for practice if time permits.
Example. Solve the simultaneous congruences
x 3 (mod 6)
x 5 (mod 35)
x 7 (mod 143)
Solution.