THEORY OF NUMBERS, Math 115 A Homework 1 Due Wednesday October 3 1. Use the well-ordering property to show that 3 is irrational. 2. Find a formula for
n k=1
k3 .
3. Find and prove a simple formula
Homework 4 solutions
July 24, 2017
4.1.2 a) Yes
b) Yes
c) No
4.1.4 If a is even we write a = 2k. Then a2 = 4k 2 , which is divisible by 4, hence a2 0
(mod 4). If a is odd write a = 2k + 1. Then a2 = 4
Math 115A
Summer 2017
Midterm
07/13/17
Time Limit: 100 Minutes
Instructor: Xiang He
Name (Print):
Student ID:
Signature:
This exam contains 5 pages (including this cover page) and 6 problems. Check to
Homework 2 solutions
July 10, 2017
3.3.6 Since (a, a + 2) = (a, 2). The gcd is 2 when a is even and 1 when a is odd.
3.3.16 a) We can find integers u, v and r, s such that ua + vb = ra + sc = 1. Hence
Homework 3 solutions
July 17, 2017
3.5.4. a) 2, 5
c) 2, 3, 5, 7
3.5.8. Let a be an integer, and write a = pa11 pann . Without loss of generality, we may assume a1 , a2 , ., am are odd and the rest are
Homework 1 solutions
July 3, 2017
1.1.2 Let S be the set of all positive integers of the form a bk. Then S is not empty
because a (1)b = a + b is a positive integer and contained in S. Then the WellOr
Homework 5 and 6 solutions
July 31, 2017
6.2.4 Let n be an odd composite integer, then 1n
( 1)n
1
( 1)2 )(n
1)/2
1
1 (mod n) and
1(n
1)/2
1
(mod n).
Hence n is a pseudoprime to the base 1.
6.2.8 By
THEORY OF NUMBERS, Math 115 A Homework 2 Due Wednesday October 12 1. Use the sieve of Eratosthenes to find all primes less than 250 2. Find all primes that are difference of the fourth powers of two i
THEORY OF NUMBERS, Math 115 A Homework 3 Due Friday October 19 1. Consider the "number system" {a + b -5 : a, b Z} (as a subset of the complex numbers. That is, -5 = 5i). It is usually denoted Z[
THEORY OF NUMBERS, Math 115 A Homework 4 Due Monday October 29 1. A student returning from Europe changes his French and Swiss francs into US. money. He only brought "whole francs", not fractions of t
THEORY OF NUMBERS, Math 115 A Homework 5 Due Friday November 9 1. Find the solutions of each of the following systems of linear congruences: x 0 x0 x 1 x6 (mod (mod (mod (mod 2) 3) 5) 7) 3x 5
THEORY OF NUMBERS, Math 115 A Homework 6 Due Monday November 19 1. What is the remainder of 7 8 9 15 16 17 23 24 25 43 modulo 11? 2. Show that the periods in the decimal expansion of 1/83 and
THEORY OF NUMBERS, Math 115 A Homework 7 Due Friday November 30 1. Show that if c1 , . . . , c(m) is a reduced system of representatives modulo any m > 2 then c1 + + c(m) 0 (mod m). 2. Use Euler's
THEORY OF NUMBERS, Math 115 A Homework 8 NOT Due Friday December 07 1. Find the sums and numbers of divisors of 196, 2100 and 10!. 2. Show that the number of divisors of n is odd if and only if n is a