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Question 1 Consider the following two Diophantine equations with integer variables X and Y (1) 99 X +...

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Advance Math

Question 1

Consider the following two Diophantine equations with integervariables X and Y

(1) 99 X + 225 Y = 36

(2) 225 X + 99 Y = 33

(a) Determine which of these two equations is inconsistent, andexplain why
(you can use the Maple commands ifactor and/or gcd ).
For the equation that is consistent, apply Extended Euclid'sAlgorithm to find a solution  {X0, Y0}
.

(b) Find now all solutions  {X, Y} of that equation,and identify the solution with smallest possible absolute valueof  Y
.  

QUESTION 2

Consider the ring of congruence classes R=?/15?:

(a) Identify the set S of all zero divisors and the set U of allunits in R. Explain your solution.

(b) For the element [10], find all complementary zero divisorsor all inverses, whichever exists. Explain.

(c) Find all solutions of the equation [23] X = [12]
if it is consistent, or explain why this equation isinconsistent.

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