Division Algorithm:
Given integers a and b with b > 0 we can nd integers q and r with
0 r < b and
a = bq + r.
Euclids algorithm for constructing greatest common divisor of a
and b = 0:
Form the sequence a0 , a1 , a2 , . . . with a1 > a2 > 0 via a0 = a,
a1
Exam 5 Solutions Math 290 Spring 2014
Name:
Provide brief and logically correct solutions to the following problems:
1. Partition the set S = cfw_x Z : 2 x 15 into seven 2-element subsets
of the form cfw_a, b where a < b and ab 1 mod 17.
Solution: One way
Exam 3 Solutions Math 290 Spring 2014
Name:
Instructions: Give brief and logically correct proofs. Write one answer per
blank sheet of paper.
1. An easy warm-up problem: nd the coecient of x10 y 20 in (2x + 3y)30 .
Solution:
30
30
(2x + 3y)
=
k=0
30
(2x)k
Exam 2 Math 290 Spring 2014
Name:
Instructions: Give brief and logically correct proofs. Write one answer per
blank sheet of paper.
1. Let A, B, and C be sets. Prove that A (B C) = (A B) (A C).
Solution 1: Using a Venn Diagram, represent A by regions cfw_
Exam 4 Solutions Math 290 Spring 2014
Name:
Instructions: All variables are integers. Make sure to check your work.
1. Find q and r so that 300 = 17q + r where 0 r < 17.
Solution: q = 18 and r = 6.
1
2. Using Euclids Algorithm, prove that gcd(23856, 23847
Exam 1 Math 290 Spring 2014
Name:
Instructions: Give brief and logically correct proofs. Write one answer per
blank sheet of paper.
Denition 1: Let a and b be integers. Then a|b means that
or equivalently b = ka for some integer k.
b
a
is an integer,
Deni