Solutions
Math 4400
HOMEWORK #4
1. If d divides n, prove that (d) divides (n). (Here, is the Euler -function.)
Let d be a divisor of n and let
d = pe1 pe2 per
r
12
be its unique factorization into powers of distinct primes. Since d divides n, then
n = pf1
Solutions
Math 4400
HOMEWORK #5
1. Consider the eld F = Z 19Z.
(a) Compute all powers 2k for k = 1, . . . , 18 to see that the order of 2 is 18.
Here are the powers of 2 modulo 19.
k
2k
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
2 4 8 16 13 7 14 9 18 17
Solutions
Math 4400
EXAM #1
1. Find all solutions in Z 12Z = cfw_0, 1, 2, . . . , 11 to the congruence x10 1 (12).
First, if x10 = 1 in Z 12Z, then x9 is the inverse of x. Therefore, any solution to this
congruence must actually be in Z 12Z . Since (12) =
Solutions
Math 4400
HOMEWORK #1
1. Let m be the least common multiple of two positive integers a, b. If M is any other common
multiple of a and b, prove that m divides M . (Hint: use division with remainder of M by m.)
Since m is the least common multiple
Numbers, Groups
and
Cryptography
Gordan Savin
Contents
Chapter 1. Euclidean Algorithm
1. Euclidean Algorithm
2. Fundamental Theorem of Arithmetic
3. Uniqueness of Factorization
4. Eciency of the Euclidean Algorithm
5
5
9
13
16
Chapter 2. Groups and Arithm
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M4400 Exam 1 Name UID
1. Suppose L is a subset of the integers Z, and L satises the following 3 properties:
(i) L is closed under addition, meaning
TyEL=>$+yEL
(ii) L is closed under multiplication by integers, meaning
TELtnez => nmeL
(iii)
M4400 Exam2 Name gals-MS \ UID
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(i) Fill in the multiplication table for 2/52
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(ii) In the modular numbe
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Solutions
Math 4400
HOMEWORK #3
1. Let G be a group. Prove that the identity element is unique. (Hint: suppose there are two
identity elements e and e , and then show they are equal by considering the product ee .)
Suppose that e and e are both identity e
Solutions
Math 4400
HOMEWORK #2
1. Let a and b be two positive integers such that a + b is a prime number. Prove that gcd(a, b) = 1.
Let d = gcd(a, b) and p = a + b a prime. Since d divides a and d divides b, it must be
true that d divides a + b = p. Sinc
Due Friday, August 2nd
Math 4400
EXAM #3
This is a take-home exam. You may use anything from class or the course notes. Do not collaborate
with anyone and do not use any additional resources. You may use computers or calculators to
perform your computatio
Math 4400
EXAM #2
Do exactly 5 of the following 6 problems on the blank paper provided. Show your work.
1. Every ring is a eld, but some elds are not rings. For each of the following rings, determine
whether or not it is a eld. For each ring that is not a
Solutions
Math 4400
EXAM #2
Do exactly 5 of the following 6 problems on the blank paper provided. Show your work.
1. For each of the following rings, determine whether or not it is a eld. For each ring that is not
a eld, demonstrate a eld property that fa
Solutions
Math 4400
REVIEW #2
1. A eld is a special type of ring. What additional conditions must a general ring satisfy in order
to be a eld? For each of these additional conditions, give an example of a ring which does not
satisfy the condition.
The axi
Math 4400
EXAM #1
Do exactly 6 of the following 7 problems on the blank paper provided. Show your work.
1. Find all solutions in Z 12Z = cfw_0, 1, 2, . . . , 11 to the congruence
x10 1 (12).
2. Find all solutions in Z 2100Z to the following system of cong
Solutions
Math 4400
REVIEW #1
1. Let d be the greatest common divisor of 1326 and 1254.
(a) Use the Euclidean algorithm to compute d.
1326 = 1 1254 + 72
1254 = 17 72 + 30
72 = 2 30 + 12
30 = 2 12 + 6
12 = 2 6 + 0
so d = gcd(1326, 1254) = 6 .
(b) Use part
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Solutions
Math 4400
HOMEWORK #6
1. Use properties of the Legendre symbol and quadratic reciprocity to determine whether 52 and
110 are squares modulo 127. Explain what this implies about the rings
and Z 127Z 110 .
Z 127Z 52
52
127
4
127
=
13
127
13
5
= (1