1. SOLVING LINEAR DIOPHANTINE EQUATIONS

2. WHAT IS A LINEAR DIOPHANTINE EQUATION?  A Diophantine equation is a polynomial equation whose solutions are restricted to integers. These types of equations are named after the ancient Greek mathematician Diophantus.polynomial equationintegers  A linear Diophantine equation is a first-degree equation of this type. Diophantine equations are important when a problem requires a solution in whole amounts.first-degree equation  We can write ax + by = c where a, b, c ∈ Z.

3. METHOD FOR COMPUTING THE INITIAL SOLUTION TO A LINEAR DIOPHANTINE EQUATION IN 2 VARIABLES

4. FIND AN INITIAL INTEGER SOLUTION TO THE EQUATION 141x+34y = 30 Using Euclidean Algorithm, Therefore gcd(141,34)=1. Solutions exist because 1 ∣ 30. Reformat the equation from the Euclidean Algorithm,

5. 141x+34y = 30

6. NOT ALL LINEAR DIOPHANTINE EQUATIONS HAVE A SOLUTION

7. Example: Find a solution to the Diophantine equation 47x + 30y = 1 The strategy is to use the Euclidean Algorithm Suppose a and b are integers. Mathematically, a ∈ Z and b ∈ Z. Then the Diophantine equation ax + by = gcd (a, b) has a solution

8. Next, solve for the remainders Use Berzout’s Theorem or Backward Substitution So the equation 47x + 30y = 1 has a solution x = -7, y = 11