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

Question 1

Consider the following two Diophantine equations with integer variables X and Y

(1) 99 X + 225 Y = 36

(2) 225 X + 99 Y = 33

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

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

QUESTION 2

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

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

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

Solutions

Expert Solution

Colby Messinger answered 1 year ago

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