Math 42-Number Theory
Problem Set #8
Due Tuesday, April 19, 2011
5. Prove that if p and q are distinct odd primes, then
p1
2
k=1
kq
+
p
q 1
2
=1
p1 q1
p
=
.
q
2
2
Solution: Consider the rectangle with
Math 42-Number Theory
Problem Set #4
Due Tuesday, March 8, 2011
1. Find all x in Z that satisfy x 14 mod 65 and x 25 mod 93.
2. Find all x in Z that satisfy x 14 mod 65 and x 25 mod 91.
3. Find all x
Math 42-Number Theory
Problem Set #2
Due Tuesday, February 15, 2011
1. Which of the following can you nd in Z7 ? (If you can nd them, give a value or values.)
12
3
4 5, 2, , , 2, 1, 6
25
2. Which of
Math 42-Number Theory
Problem Set #8
Due Tuesday, April 19, 2011
1. Compute
5
k=1
5k
.
11
Use this to determine whether 5 is a square mod 11 or not.
2. Compute
6
k=1
5k
.
13
Use this to determine whet
Math 42-Number Theory
Problem of the Day #1
Due Tuesday, February 1, 2011
1. In a game where you can score either a or b points at a time, where a and b are in N and
relatively prime, what is the maxi
Math 42-Number Theory
Problem Set #1
Due Tuesday, February 8, 2011
1. Use Euclids algorithm to nd the GCD of 29 and 11.
2. Write
29
11
as a simple continued fraction. Compute all the convergents.
3. F
Math 42-Number Theory
Problem Set #5
Due Thursday, March 17, 2011
1. Find a generator for U29 . Use it to make a table of logarithms. Use your table of logarithms
from problem 1 to solve 13x3 = 21 mod
Math 42-Number Theory
Problem Set #7
Due Thursday, April 7, 2011
1. Factor 30 in Z[i]. Is there an element of Z[i] with norm 30?
2. Factor 65 in Z[i]. Is there an element of Z[i] with norm 65?
3. How
Math 42-Number Theory
Problem Set #6
Due Thursday, March 24, 2011
1. List the squares (quadratic residues) in U19 , in U23 and in U29 . How many squares are there in
Up ?
2. Give a characterization of
Math 42-Number Theory
Problem Set #7
Due Thursday, April 7, 2011
10. Prove that given natural numbers a and b, there exist integers q , r, such that a = bq + r
2a
b
where = 1 and 0 r 2 . Prove in addi
Math 42-Number Theory
Problem Set #6
Due Thursday, March 24, 2011
2. Give a characterization of the squares in terms of a generator. That is, which powers of a
generator are squares?
Solution: Even po
M ATH 42 P RACTICE M IDTERM 2
Name:
F OR F ULL C REDIT
S HOW A LL W ORK
N O C ALCULATORS
1. There is a marching band getting into certain congurations. When they line up in lines of 7, there is one
pe
Math 42-Number Theory
Problem Set #1
Due Tuesday, February 8, 2011
8. In a game where you can score a or b points at a time (a, b N), what is the largest unattainable
score? Explain why this score is
Math 42-Number Theory
Problem Set #2
Due Tuesday, February 15, 2011
For problems 4, 5 and 6, either prove the statement is true or give an example showing that it is
false (a counterexample).
4. If a
Math 42-Number Theory
Problem Set #3
Due Thursday, February 24, 2011
2. Prove that if a b mod m and c d mod m, then ac bd mod m.
Solution: If a b mod m and c d mod m, then by denition, m | (b a) and m
Math 42-Number Theory
Problem Set #4
Due Tuesday, March 8, 2011
8. Prove that if g is a generator for Up , then g k has order
prime.
p1
d,
where d = (p 1, k ). Here, p is
Solution: Suppose (g k )x 1 m
Math 42-Number Theory
Problem Set #5
Due Thursday, March 17, 2011
7. Prove that if u1 and u2 are elements of Um with orders n1 and n2 respectively and (n1 , n2 ) = 1,
then the order of u1 u2 is n1 n2
Math 42-Number Theory
Problem Set #3
Due Thursday, February 24, 2011
1. List the squares in Z5 , Z7 , Z11 and Z13 . How many squares are there in each case? Make a
general statement about the number o