Exercise 3.13. For each of the following equations, decide...

Exercise 3.13. For each of the following equations, decide whether it can be solved in integers. If so, find a solution: 1. $6x + 9y = 2$; 2. $6x + 9y = -33$; 3. $25x + 40y = 345$; 4. $14x + 85y = 3$; 5. $66x + 561y = 22$; 6. $66x + 561y = 99$; 7. $70x + 1869y = 35$; 8. $3145x + 23001y = 4$; 9. $3145x + 23001y = -85$.

Transcript

00:01 Hello students, a diophantine equation is the type of equation in which only integer solutions are solved.

00:08 Let's go through each equation and determine whether it can be solved in integers and if so find infinitely many solutions.

00:14 The first one is 14x plus 85y equal to 3.

00:20 This equation is solvable in integers.

00:22 To find the infinitely many solution we can use the fact of equation can be rewritten as 14x plus 85y equal to 3 that is equal to 14x plus 85y minus 5 into 17 equal to 3 minus 5 into 17.

00:39 So we can write it as 14x plus 85y minus 85 equal to minus 82.

00:47 So 14x plus 85y minus 1 equal to minus 82.

00:53 Now choosing the integer value for x.

00:56 Let's say x is equal to k and solve for y.

01:00 So 85y minus 1 equal to minus 82 minus 14k.

01:06 So y minus 1 equal to minus 82 minus 14k by 85.

01:13 Since minus 82 minus 14k is always divisible by 85, we can choose different values for k to generate infinitely many solution.

01:25 Infinitely many solution.

01:27 For instance, let k taken any integer value we can find corresponding y values that satisfy the equation.

01:38 Now b option 66x plus 561y equal to 22.

01:44 This equation is solvable in integers.

01:46 To find infinitely many solution we can follow similar approach.

01:50 So 66x plus 561y minus 17 into 33 that is equal to 22 minus 17 into 33.

02:00 We can rewrite as 66x plus 561y minus 1 equal to minus 539...

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