Dr. Marques Sophie
Oce 519
Number theory
Fall Semester 2013
marques@cims.nyu.edu
Problem Set #2
Due monday 16 September in Class
Exercise 1: ( )
Compute gcd(935, 1122) and gcd(1876, 4534) by using Euclidean algorithm.
Solution:
Euclidean algorithm equatio

Dr. Marques Sophie
Oce 519
Number theory
Fall Semester 2013
marques@cims.nyu.edu
Problem Set #1 with solution
Due monday 16 September in Class
Exercise 1:
Let a Z . Prove that gcd(2a + 3, a + 2) = 1.
Solution:
We have
gcd(2a+3, a+2) = gcd(a+1+a+2, a+2) =

Dr. Marques Sophie
Oce 519
Number theory
Fall Semester 2013
marques@cims.nyu.edu
Problem Set #11
Due monday December 2nd in Class
Exercise 1: ( ) 4 points
Find the minimal solutions to x2 3y 2 = 1, x2 6y 2 = 1.
Solution:
Let d be a positive square free in

Dr. Marques Sophie
Oce 519
Number theory
Fall Semester 2013
marques@cims.nyu.edu
Problem Set #8
Due monday november 4th in Class
Exercise 1: ( ) 4 points
Let be Eulers function. Show that (nm ) = nm1 (n) for all natural number n, m.
Solution:
Let n = pa1

Dr. Marques Sophie
Oce 519
Number theory
Fall Semester 2013
marques@cims.nyu.edu
Problem Set #10
Due monday november 25th in Class
Exercise 1: ( ) 4 points
Show that if p 7 is an odd prime, then
n
n+1
=
p
p
for at least one of n = 2, 4, 5, or 8.
Solution:

Dr. Marques Sophie
Oce 519
Number theory
Fall Semester 2013
marques@cims.nyu.edu
Problem Set #9
Due monday november 11th in Class
Exercise 1: ( ) 3 points
3
17
Compute 11 , 10 , and 101 .
17
Solution:
By using quadratic residue, we can easily compute them

Dr. Marques Sophie
Oce 519
Number theory
Fall Semester 2013
marques@cims.nyu.edu
Problem Set #7
Due monday october 28th in Class
Exercise 1: ( ) 4 points
Obtain three consecutive integers, each having a square factor.
Solution:
The idea here is to set up

Dr. Marques Sophie
Oce 519
Number theory
Fall Semester 2013
marques@cims.nyu.edu
Problem Set #3
Due monday 30 September in Class
Exercise 1: ( ) 2 points
Find all solutions of the conguence 12x 3 mod 45.
Solution:
Since gcd(12, 45) = 3 and 3 0 mod 3, the

Dr. Marques Sophie
Oce 519
Number theory
Fall Semester 2013
marques@cims.nyu.edu
Problem Set #5
Due monday october 21th in Class
Exercise 1: ( ) 4 points
Prove that there are innitely many integers which can not be written as a sum of three
squares. (Hint

Dr. Marques Sophie
Oce 519
Number theory
Fall Semester 2013
marques@cims.nyu.edu
Problem Set #3
Due monday october 7th in Class
Exercise 1: ( ) 4 points
Find a solution of the congruence 5x3 93 mod 231. (Hint: Use CRT.)
Solution:
First, note that 231 = 3