Summer 2008 COT 3100 Homework #2
Given: 5/29/08
Due: 6/6/08 (in recitation)
6) Prove that
p is irrational, where p is any prime number.
7) When dividing integer a by integer b, we write a = bq + r, where q and r are integers
such that 0 ≤ r < b. Prove that if a > b, then a > 2r.
8) Use the Euclidean Algorithm to determine the GCD of 1365 and 286.
9) Use the Extended Euclidean Algorithm to find integers x and y that satisfy the
following equation: 1365x + 286y = gcd(1365, 286).
10) Prove that the equation 323x + 204y = 1000000 has no integer solutions for x and y.
Optional Programming Homework
If you do this question, you don't have to do questions 8, 9 and 10.
Write a program that solves equations of the form Ax + By = C, where A, B and C are
given positive integers. Your program should output all non-negative integer solutions for
x and y. If there are no solutions, then output a single line saying so.
Note: This program has a practical application in that a common problem people must
solve is of the following type: Each object of type one costs A dollars and each object of
type two costs B dollars. I must spend exactly C dollars. What are my different options
for satisfying these constraints?
Sample Run A
Please enter A, B and C.
6 8 100
Here are all the solutions:
x = 2 y = 11
x = 6 y = 8
x = 10 y = 5
x = 14 y = 2
Sample Run B
Please enter A, B and C.
12 17 100
This equation has no solutions.