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 all Pythagorean triples (a, b, c) with a and b consecuti
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 full
credit. Please note that the handouts on Divisibility
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, 6, 7, 8, 9, 10, 11, 12, 13, 14 .
Here is the standard
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 satisfies the congruence b 3 (mod 19). What can you say abo
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 x such
that x gives a remainder of 1 when di
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). Notice that 23 11 (mod 17)
because 23 11 = 34 is divisible
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 class one day) implies that (c, a) = (a, b). It is
obvious
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 definition,
F1 = 1, F2 = 1, and Fn = Fn1 + Fn2 for all n 3. Pr
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. Are there solutions x such
that x gives a remainder of 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 is it true that (2n) > (n)?
5. Let p be an odd prime and
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
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: Use the proof of Eisensteins Lemma.
3. How many quadr
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 is a
primitive root mod p then a is a primitive root mo
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 primitive root
mod p2 if a p1 1 mod p2 . To do this ques
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. Let p be an odd prime and a be an integer such that 1
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.
Solution: Recall that a has order r mod p if ar 1 mod p a
1
Homework 2
1. Let n N and n
1. Show that 2n 1 is not a perfect square.
2. Let n N and n
2. Suppose n = pq where p, q are distinct
Mersenne primes. Show that (n) is equal to a power of 2.
3. Show that an + 1 is composite if n has an odd divisor.
4. Show
1
Homework 2
1. Let n N and n > 1. Show that 2n 1 is not a perfect square.
Solution: Suppose 2n 1 = m2 for some m N. Then m must be odd,
so write m = 2k + 1 for some k N. This gives 2n 1 = (2k + 1)2 so
2n 2 = 4k2 + 4k
Since n > 1, we can write 2n1 1 = 2(k
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 + 4 + 8 + 2 + 9 + 1 9 + 11 + 12 + 12 26 1
(mod 9) .
We ha
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 side just has one term, namely 13 = 1 and so the left si
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 the converse of (a) true? i.e. Is it true that if 2”‘1
Math 301A
Spring 2016
\
X
\
\¥\
H
H
$995?
7“
CD
\ \5\
\
. Use the previous problem to solve the congruence: 73: E 3 mod 30. Verify the answer is correct. / ,1,
Homework 4
Dudley p.48: 1 ~59? baa'L
Dudley p.48: 2 -' c5!
Dudley p.48: 3 v S‘eC-
Dudley p.48
Homework 9
1. Read the pdf sent out this week, especially the section related to the
Fermat-Pell equations.
2. Evaluate the continued fraction 2, 1, 3, 2, 4 .
3. Expand 6 as a continued fraction.
4. Give two solutions (not counting x = 1, y = 0) to x2 6y
(1) Let a be the number given by the rst 5 digits of your student
id number. Let b be the number given by the last 4 digits of
your student id number. Find (a, b) and express it as a linear
combination of a and b.
The answer will be dierent for every stud
For the nal you should be familiar with the following sections of the
book: 1.2, 2.1-2.5, 3.1-3.7, 5.1,5.2,the main theorem of 5.3,5.4,8.2-8.3
(at least the concepts touched upon in class) and the continued fractions material (from lecture notes, but read
Homework 1 Answers
1. (Pg 14 # 4) Assume a|b and b|a. By denition this means there exists
x such that ax = b and y such that by = a. Substituting ax for b in
the second equation we get axy = a. Since a = 0 (which is an implicit
assumption in saying a|b) w