Math 312. Assignment 8 solutions.
Problem 7.2.4 Let n = 2d pd1 . . . pdr , where d 0, d1 , . . . , dr 1 and p1 , . . . , pr
r
1
are odd primes. Then
(n) = (2d )(pd1 ) . . . (pdr )
r
1
is odd, if and o
PRACTICE SET 1
Section 1.3
1
1
1
1
Exercise 4. Conjecture a formula for nk=1 k(k+1)
= 12
+ 23
+ + n(n+1)
from the value of this
sum for small integers n. Prove that your conjecture is correct using ma
PRACTICE SET 3
Section 6.1
Exercise 4. What is the remainder when 5!25! is divided by 31?
Exercise 10. What is the remainder when 62000 is divided by 11?
Exercise 12. Using Fermats little theorem, fin
University of British Columbia
Math 312, Midterm, 30th Sep 2016
Version A
Name (please be legible)
Signature
Student number
INSTRUCTIONS
Duration: 50 minutes
This test has 4 problems for a total of
PRACTICE SET 2
Section 4.1
Exercise 4. Show that if a is an even integer, then a2 0 (mod 4), and if a is an odd
integer, then a2 1 (mod 4).
Exercise 30. Show by mathematical induction that if n is a p
PROBLEM SET A
Exercise 1. This exercise develops a number system that does not possess unique factorization and thus has no Fundamental Theorem of Arithmetic.
Let Z[ 5] denote the setof all complex nu
COMMENTS ON ASSIGNMENT 4
1. Comments from the marker
Some solutions still look quite similar to each other. Remember, that you may
talk about the problems together but you are responsible for writing
Math 312. Assignment 6 solutions.
Problem 5.1.4. Since 5 26 (mod 31), 4 27 (mod 31), etc., we have
5! 25! 25! (26) (27) (28) (29) (30) (1)5 30!
(mod 31)
5
By Wilsont theorem, the last expression equal
COMMENTS ON ASSIGNMENT 1
General comments
Handwriting. Many papers are hard to read because of bad handwriting.
Please try to write clearly and keep your notations consistent.
Explain your solutions.
Math 312. Assignment 5 solution outlines.
Problem 4.3.2. Need to solve the system of congruences
x 1 (mod 2)
x 0 (mod 3)
x 1 (mod 5)
for x. Note that 2, 3 and 5 are pairwise relatively prime. The form
COMMENTS ON ASSIGNMENT 2
General: Once again, presentation is important. Please solve the problem and
clean up your solution before writing it down. Indicate intermediate steps and
leave out observati
COMMENTS ON ASSIGNMENT 3
General comments
The marker tell me that some of the HW3 papers are virtually identical to
each other, down to the notation and the wording. We assumed this resulted from
misu
Math 312. Assignment 9 solutions.
Problem 8.1.2 Answer: I CAME I SAW I CONQUERED
Problem 8.1.4 Answer: NPWJEAPNSPQESW
Problem 8.1.6 Solving C 3P + 24 (mod 26) for P , we obtain 3P C + 2
and thus
P 31
Math 312. Assignment 10 solutions.
Problem 8.4.8 To break the code, we factor 2881 = 43 67. Now (2881) =
42 66 = 2772. Using the Euclidean algorithm, we compute the decryption key
d e1 51 1109 (mod 27
Math 312. Assignment 7 solutions.
Problem 6.2.18(a) Let n = p1 p2 p3 , where p1 = 6m + 1, p2 = 12m + 1 and
p3 = 18m + 1 are primes. We have to show that if gcd(a, n) = 1, then an1 1
(mod n). By the Ch
SOLUTIONS TO PROBLEM SET 1
Section 1.3
Exercise 4. We see that
1
1
1
2
1
1
1
3
1
= ,
+
= ,
+
+
= ,
12 2
12 23 3
12 23 34 4
and is reasonable to conjecture
n
1
n
=
.
n+1
k=1 k(k + 1)
We will prove this