MATH 433
February 6, 2015
Quiz 3: Solutions
Problem 1. Determine the last digit of 433433 .
Solution: 3.
The last digit is the remainder under division by 10. We have 433 3 mod 10. Then
4332 = 433 433 3 3 = 9 (mod 10),
4333 = 4332 433 9 3 = 27 7 (mod 10),
MATH 433
Spring 2015
Sample problems for Exam 1
Any problem may be altered, removed or replaced by a dierent one!
Problem 1. Find gcd(1106, 350).
Problem 2. Find an integer solution of the equation 45x + 115y = 10.
Problem 3. Prove by induction that
1
1
1
MATH 433
March 2, 2015
Quiz 5: Solutions
Problem 1. Let R be the relation dened on the set of positive integers by xRy if and only
if x = y n for some positive integer n (is a power of). Is this relation reexive? Symmetric?
Transitive? Explain how you kno
MATH 433
February 20, 2015
Exam 1: Solutions
Problem 1 (20 pts.) Find the smallest positive integer a such that the equation
76x + 96y = a has an integer solution.
Solution: a = 4.
The sought number is the greatest common divisor of 76 and 96. Since 76 =
MATH 433
February 13, 2015
Quiz 4: Solutions
Problem 1. Solve a linear congruence 4x 18 mod 11.
Solution: x 10 mod 11.
To solve this linear congruence, we need to nd the inverse of 4 modulo 11. For this, we need to represent
1 as an integral linear combin
MATH 433
Spring 2015
Sample problems for Exam 2
Any problem may be altered, removed or replaced by a dierent one!
Problem 1. Let R be a relation dened on the set of positive integers by xRy if and only
if gcd(x, y) = 1 (is not coprime with). Is this relat
MATH 433
January 30, 2015
Quiz 2: Solutions
Problem 1. Using the induction principle, prove that
for every positive integer n.
1
1
1
1
+
+ +
= 1
12 23
n(n + 1)
n+1
First consider the case n = 1. In this case the formula reduces to
Now assume that the form
MATH 433
January 26, 2015
Quiz 1: Solutions
Problem 1. Find all integers 1 x 100 such that gcd(x, 24) = 12 and gcd(x, 35) = 5.
Solution: x = 60.
Since gcd(x, 24) = 12, the number x is divisible by 12 but not divisible by 24. Hence x = 12k, where
k is an o
MATH 433
March 27, 2015
Quiz 8: Solutions
n 0
, where n and k are
k n
integers. Under the operations of matrix addition and multiplication, does this set form a ring? Does
M form a eld? Explain.
Problem 1.
Let M be the set of all 2 2 matrices of the form
MATH 433
March 13, 2015
Quiz 7: Solutions
Problem 1. Consider a binary operation on R given by x y =
Is (R, ) a group? Explain.
3
x3 + y 3 for any x, y R.
Solution: Yes, (R, ) is a group.
We need to check four axioms. First of all, the set R is closed und