Math 465 Problem Set 6
Due Friday, March 21, 2008
1. Evaluate the following Legendre symbols. (Note that 127 is prime.) (a) (b) (c) (d)
2 127 -1 127 11 127 15 127
2. How many solutions do the following congruences have? (a) x2 2 (mod 61) (b) x2
Math 465 Problem Set 10
Due Wednesday, April 30, 2008
1. Determine whether each of the following integers can be expressed as a sum of two squares. If so, find such a representation. (a) 65 (b) 95 2. List your favorite topics from this course. 3. W
Math 465 Solution Set 4
1. Solve the simultaneous congruences x3 x5 x7 (mod 6), (mod 35), (mod 143).
Ae Ja Yee
Solution. Since 1 = 6 6 + 35 (-1), we obtain x 5 6 6 + 35 (-1) 3 75 We now solve x 75 x7 (mod 210), (mod 143). (mod 210).
We ap
Math 465, Exam I (100 pts) Print Name:
February 25, 2008 Ae Ja Yee
This exam is closed-book, closed-notes. Show all your work for full credit. Partial credit will be given based on what is written. You have 50 minutes to finish the exam. 1. (15 pts
Math 465 Solution Set 5
1. Find all incongruent quadratic residues and nonresidues modulo 13. Solution. We square the first 6 integers: 12 1, 22 4, 33 9, 42 3, 52 12, 62 10.
Ae Ja Yee
Since we know there are 6 quadratic residues, they are the
Math 465, Exam I (100 pts) Print Name:
February 23, 2007 Ae Ja Yee
This exam is closed-book, closed-notes. Show all your work for full credit. Partial credit will be given based on what is written. You have 50 minutes to finish the exam. 1. (10 pts
Math 465 Solution Set 1
1. Let a, b, c Z. Prove each of the following. (a) If ac|bc and c = 0, then a|b. (b) If a|b, then ac|bc. Solution. (a) Since bc = acx for some x Z and c = 0, b = ax. (e) b = ax for some x Z, so bc = acx. 2. Prove that if n
MATH 465 NUMBER THEORY, SPRING
TERM 2013, PRACTICE EXAM 1.
Note: Exam 1 will be 1:252:15, Wednesday 13th February 2013
Room 075 Willard
1. (25 marks) Suppose that l, m, n N. Prove that (lm, ln) = l(m, n).
2. (25 marks) (i) Show that if (l, 6) = 1, then l
MATH 465 NUMBER THEORY, SPRING
TERM 2013, PRACTICE EXAM 2.
Note: Exam 2 will be 1:252:15, Wednesday 20th March 2013
Room 075, Willard
1. (25 marks) Solve the simultaneous congruences
x 4 (mod 19)
x 5 (mod 31)
2. (25 marks) Find all solutions to the congru
MATH 465 NUMBER THEORY, SPRING TERM
2013, PRACTICE EXAM 1, MODEL SOLUTIONS
Note: Exam 1 will be 1:252:15, Wednesday 13th February 2013
Room 075 Willard
1. Suppose that l, m, n N. Prove that (lm, ln) = l(m, n).
Suppose that m, n, l have the canonical decom
MATH 465 NUMBER THEORY, SPRING TERM
2013, PRACTICE EXAM 2, MODEL SOLUTIONS
1. (25 marks) Solve the simultaneous congruences x 4 (mod 19), x 5 (mod 31).
Solve 31a 1 (mod 19) and 19b 1 (mod 31). By Euclids algorithm, 1 = 8.31
13.19, Thus a = 8, b 13 18 (mo
MATH 465 NUMBER THEORY, SPRING
TERM 2013, PRACTISE FINAL
Note: The Final Exam for Math 465 will be
Wednesday 1st May, 4:40pm6:30pm in room 075 Willard.
1. Find (1745, 1485) and integers x and y such that 1745x + 1485y = (1745, 1485).
2. Let x and y be int
Math 465 Number Theory, Spring 2013, Practise Final, Solutions
Note: The Final Exam for Math 465 will be
Wednesday 1st May, 4:40pm6:30pm in room 075 Willard.
1. Find (1745, 1485) and integers x and y such that 1745x + 1485y = (1745, 1485).
(1745, 1485) =
MATH 465 NUMBER THEORY, SPRING 2013, PROBLEMS 10
Return by Monday 25th March
Complex numbers and the exponential function
This homework is not connected with the more recent classwork, but does use
some
theory of divisiblity and residue classes. Let i de
Math 465 Number Theory, Spring 2013, Practise Final, Solutions
Note: The Final Exam for Math 465 will be
Wednesday 1st May, 4:40pm6:30pm in room 075 Willard.
1. Find (1745, 1485) and integers x and y such that 1745x + 1485y = (1745, 1485).
(1745, 1485) =
MATH 465 NUMBER THEORY, SPRING 2013, PROBLEMS 13
Return by Monday 15th April
1. (i) Prove that 2 is a quadratic residue modulo p i p 1 or 3 mod 8.
Henceforth suppose p 1 or 3 mod 8.
(ii) Deduce that there are x Z, y Z, not both 0, such that |x| < p, |y |
Math 465 Solution Set 9
Ae Ja Yee
1. Prove that {3, 4, 5} is the only primitive Pythagorean triple involving consecutive positive integers. Solution. Let x = a - 1, y = a, z = a + 1. Then (a - 1)2 + a2 = (a + 1)2 2a2 - 2a + 1 = a2 + 2a + 1 a2 - 4a
Math 465 Solution Set 2
1. If (a, c) = 1 and (b, c) = 1, then (ab, c) = 1.
Ae Ja Yee
Solution. Method 1: Suppose (ab, c) > 1, namely x | ab and x | c for some x > 1. Since (a, c) = 1, x cannot divide a, that is (x, a) = 1. However, x | ab, so x | b
Math 465 Solution Set 8
1. Find all integer solutions to the Diophantine equation 40x + 25y = 600. Solution. The equation is equivalent to 40x 600 8x 120 3x 0 Thus x0 or equivalently x = 5k, y = 24 - 8k (mod 5), (mod 25), (mod 5),
Ae Ja Yee
(mod
Math 465 Solution Set 3
Ae Ja Yee
1. Prove that if a and b are integers of the form 6n + 1 or 6n + 3, then the product ab is also of the form 6n + 1 or 6n + 3. Solution. ab = (6m + 1)(6n + 1) = 6(6mn + m + n) + 1, ab = (6m + 1)(6n + 3) = 6(6mn + 3m
MATH 465, Exam II (100 pts) Print Name:
April 6, 2007 Ae Ja Yee
This exam is closed-book, closed-notes. Show all your work for full credit unless otherwise indicated. Partial credit will be given based on what is written. You have 50 minutes to fin
Math 465, Exam II (100 pts) Print Name:
April 7, 2008 Ae Ja Yee
This exam is closed-book, closed-notes. Show all your work for full credit unless otherwise indicated. Partial credit will be given based on what is written. You have 50 minutes to fin
Math 465, Final (150 pts) Print Name:
May 5, 2008 Ae Ja Yee
This exam is closed-book, closed-notes. Show all your work for full credit unless otherwise indicated. Partial credit will be given based on what is written. You have 110 minutes to finish
Math 465, Final (150 pts) Print Name:
May 9, 2007 Ae Ja Yee
This exam is closed-book, closed-notes. Show all your work for full credit unless otherwise indicated. Partial credit will be given based on what is written. You have 110 minutes to finish
Math 465, Sample Exam III SHOW ALL WORK. INDICATE ALL REASONING. 1. Go over all the problems on the midterms, sample exams, and homework. 2. Find all integer solutions to the linear Diophantine equation 252x + 66y = 6000. 3. Find all primitive Pythag
Math 465 Problem Set 3
Due Friday, February 8, 2008
1. Prove that if a and b are integers of the form 6n + 1 or 6n + 3, then the product ab is also of the form 6n + 1 or 6n + 3. 2. Prove that there are infinitely many primes of the form 6n + 5. 3.
Math 465 Problem Set 2
1. If (a, c) = 1 and (b, c) = 1, then (ab, c) = 1.
Due Friday, February 1, 2008
2. Show that if (a, b) = 1, then (a - b, a + b) = 1 or 2. When is the value 2? 3. Let a, b, c Z with a = 0 and b = 0. Show that if (a, b) = 1, a
Math 465 Solution Set 10
Ae Ja Yee
1. Determine whether each of the following integers can be expressed as a sum of two squares. If so, find such a representation. (a) 65 (b) 95 Solution. (a) 65 = 5 13 = (12 + 22 )(22 + 32 ) = (1 2 + 2 3)2 + (1