PROBLEM SET 4 (due on Friday, May 8th)
A. A certain integer a gives a remainder of 1 when divided by 2. What can you say about
the remainder that a gives when divided by 8?
B. A certain integer b sati
SOLUTIONS FOR PROBLEM SET 5
A: Give a complete residue system modulo 15 and a reduced residue system modulo 15.
SOLUTION. Here is the standard complete residue systems modulo 15:
cfw_0, 1, 2, 3, 4, 5,
FINAL EXAM FOR MATH 301
INSTRUCTIONS: Please read the questions carefully. All of your answers should be fully
justied. Your justications and proofs must be convincing and clearly presented to get ful
1
Homework 3
1. Let p be a prime and a, b N with 0 < a, b < p. Assume that
gcd(r, s) = 1. If the order of a is equal to r and the order of b is equal
to s, show that the order of ab is equal to rs.
So
SOLUTIONS FOR THE FINAL EXAM FOR MATH 301, FALL 2014
QUESTION 1.
(a) Find the remainder when 185648291 is divided by 99.
SOLUTION: We use Casting Out Nines, to conclude that
185648291 1 + 8 + 5 + 6 +
SOLUTIONS FOR PROBLEM SET 1
A. Do parts (b), (c), and (d) of problem 1 on pages 13-14.
(b) SOLUTION: The identity in question is
n
k3 =
P (n) :
k=1
n2 (n + 1)2
4
.
To verify P(1), note that the left s
NUMBER THEORY PROBLEM SET 2 (due Wednesday, April 15th)
A: There exist primes p such that p + 6k is also prime for k = 1, 2 and 3. One such prime
is p = 11. Another such prime is p = 41. Prove that th
SOLUTIONS FOR PROBLEM SET 2
A: There exist primes p such that p + 6k is also prime for k = 1, 2 and 3. One such prime
is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime
Some Possible Questions for the Final Exam
Question 1. Find all solutions to the congruence 13x 12 (mod 35). Also, answer the
following questions about the solutions to the above congruence. Are there
SOLUTIONS FOR THE MIDTERM (QUESTIONS 1 - 4)
QUESTION 1. Suppose that a Z and that a 23 (mod 17).
(a) Find the remainder that a gives when divided by 17.
SOLUTION. We are given that a 23 (mod 17). Noti
SOLUTIONS FOR PROBLEM SET 3
A: Do problem 2 on page 25.
SOLUTION: In this problem, it is assumed that a, b, c, and k are integers. Let
c = b + ka = ka + b .
The See-Saw Lemma (which was proved in clas
1
Homework 3
1. Let p be a prime and a, b N with 0 < a, b < p. Assume that
gcd(r, s) = 1. If the order of a is equal to r and the order of b is equal
to s, show that the order of ab is equal to rs.
2.
1
Homework 4
1. Find an example of ( p, a) where p is an odd prime and a is a primitive
root mod p, but a is not a primitive root mod p2 .
Solution: Let a be a primitive root mod p, then it is not a p
1
Homework 4
1. Find an example of ( p, a) where p is an odd prime and a is a primitive
root mod p, but a is not a primitive root mod p2 .
2. Let be an integer
2 and p is an odd prime. Show that if a
1
Homework 5
1. Let p be an odd prime. Show that if a is a quadratic residue mod p
for some > 0, then a is a quadratic residue mod p+1 .
2. Let p be an odd prime, show that
2
p
= (1)( p
2 1) /8
.
Hint
Math 301. Summer Quater 2015. Midterm.
Number of questions: 5
Total marks: 25
Time allowed: 55 minutes
No calculators allowed
You may not remove the test paper from the room
Question 1 . . . . .
1
Review Questions
1. What are the last two digits of 21000 and 31000 ?
2. Solve the congruence x3 + 4x + 8 0 mod 15.
3. Prove that there are innitely many integers such that 3 | (n).
4. For which n i
Possible Questions for the Final Exam - Solutions
Question 1. Find all solutions to the congruence 13x 12 (mod 35). Also, answer the
following questions about the solutions to the above congruence. Ar
NUMBER THEORY PROBLEM SET 1 (due Wednesday, April 8th)
A. Do parts (b), (c), and (d) of problem 1 on pages 13-14.
B. This problem concerns the Fibonacci sequence cfw_Fn defined on page 11. By definit
NUMBER THEORY PROBLEM SET 3 (due Wednesday, April 22nd)
A: Do problem 2 on page 25.
B: Do problem 8(b) on page 25.
C: Do problem 9 on page 25.
D: Suppose that a and b are integers, not both zero. Let
Math 301A
Spring 2016
Homework 2
1. Dudley p.112: 1
421' Dudley p.113: 15
/3. Dudley p.113: 16
/4. Dudley p.19: 1
/\ Dudley p.19: 6
/6. (a) Prove that if n is composite, then 2"‘1 is composite.
(b) Is
Math 301A ' -
Spring 2016 ‘
Homework 3
'. The most well known Pythagorean triples are (3, 4, 5) and (5, 12, 13) and have the property that two sides are consecutive.
Consider the problem of ﬁnding a
Homework 6 Solutions
Math 301
Exercise 5.1:
pn
Claim: If cn = pqnn is the n th convergent of [a0 , a1 , ., an ] and a0 > 0, [an , an1 , ., a1 , a0 ] = pn1
Proof: Base Case: Let n = 0 then [a0 ] = a0 =
Homework # 4 Solutions
Math 301 Spring 2016
Exercise 2.23: Find all four solutions to the equation x2 1 0 (mod 35).
Note this is equivalent to x2 1 (mod 35).
Sage:
x for x in range(0, 35) if x*2 % 35
Homework # 5 Solutions
Math 301 Spring 2016
Exercise 3.1: This problem concerns encoding phrases using numbers using the encoding of
Section 3.3.2. What is the longest that an arbitrary sequence of le
Homework 2 Solutions
Math 301 Spring 2016
Exercise 1.9 a, b, c, n Z
(a) Claim: If a | b and b | c then a | c.
Proof: Suppose a | b then b = ak for some k Z. Suppose b | c then c = bl for some l Z.
The
Homework 3 Solutions
Math 301 Spring 2016
Exercise 2.4:
Claim: If a and b are integers and p is prime, then (a + b)p ap + bp (mod p).
p!
Note: You may assume that the binomial coefficient k!(pk)!
is a