Consider the following two diophantine equations. For each one, explain whether or not it has a solution. If there are any solutions, give three different solutions. (a) 19x + 22y = 6 (b) 22x - 55y = 14

Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter2: Working With Real Numbers
Section2.7: Problem Solving: Consecutive Integers
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Question 1
Consider the following two diophantine
equations. For each one, explain whether or
not it has a solution. If there are any solutions,
give three different solutions.
(a) 19x + 22y = 6
(b) 22x - 55y
14
=
Question 2
Consider the following three congruences:
x = 1 (mod 3)
x = 3 (mod 5)
x = 2 (mod 7)
(a) Find an integer x that satisfies all three
congruences simultaneously. Use the
Pulverizer, and the Chinese Remainder
Theorem, and explain how you reached your
solution, step by step.
(b) What solutions are there such that 0 ≤ x ≤
200?
Transcribed Image Text:Question 1 Consider the following two diophantine equations. For each one, explain whether or not it has a solution. If there are any solutions, give three different solutions. (a) 19x + 22y = 6 (b) 22x - 55y 14 = Question 2 Consider the following three congruences: x = 1 (mod 3) x = 3 (mod 5) x = 2 (mod 7) (a) Find an integer x that satisfies all three congruences simultaneously. Use the Pulverizer, and the Chinese Remainder Theorem, and explain how you reached your solution, step by step. (b) What solutions are there such that 0 ≤ x ≤ 200?
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