Math 365
Homework #8 due Nov. 2
Fall 2007
book problems
8.1 # 4. Let r be the order of ab (mod n). By Thm 8.1 to show r | hk
it is enough to show that (ab)hk 1 (mod n). But this is clear since
(ab)hk = (ah )k (bk )h 1 (mod n). To see that if gcd(h, k) = 1
Math 365
Exam 1 study guide
Fall 2007
I. Be able to state:
1. Well Ordering Principle
2. Binomial Theorem
3. Division Algorithm (2.1)
4. GCD Theorem (2.3)
5. Relatively Prime Theorem (2.4)
6. Fundamental Theorem of Arithmetic & canonical form corollary (3
Selected solutions, Homework 3
Math 365
Fall 2007
book problems
3.2 #13(c) If gcd(m, n) = 1 then gcd(Rm , Rn ) = 1.
Let x, y Z be such that 1 = mx + ny. Then,
Z
10 1
9
10mx 10ny 1
=
9
10mx 10ny 10mx + 10mx 1
=
9
mx
= 10 Ryn + Rmx
1 =
By 13(a), Ry | Ryn an
Math 365
Select solutions, Homework 4
Fall 2007
book problems
3.3 #20 p is equal to one of 3k, 3k + 1 or 3k + 2. When p = 3 we can check
directly what happens, so we need only consider the case p = 3k for
k > 1. In this case p is not prime so we neednt ch
Math 365
Homework 2 Select solutions
Fall 2007
book problems
2.3
#18 Let k, k + 1, k + 2 be three consecutive integers. By the division
algorithm, one of them is divisible by 3, hence 3 | k(k + 1)(k + 2).
By a similar argument, 2 | k(k + 1)(k + 2). Hence
Math 365 Study guide for take home Final Exam due December 14, 12:15pm
Fall 2007
Exam 1 material
I. Be able to state:
1. Division Algorithm (2.1)
2. GCD Theorem (2.3)
3. Relatively Prime Theorem (2.4)
4. Fundamental Theorem of Arithmetic & canonical form