But first, let’s take a quick break from all the seriousness and have some fun.

What are Diophantine Equations? Well, they’re basically math problems that involve finding integer solutions to polynomial equations with two or more variables. And if you think that sounds boring, just wait until we talk about linear diophantine equations! These babies only involve one variable and a constant coefficient, making them the easiest type of Diophantine equation to solve.

But why would anyone want to solve these things? Well, for starters, they’re useful in cryptography and coding theory. And if you’re not into that stuff, maybe you just enjoy solving math problems for fun! Either way, let’s dive right in and learn about the algorithm.

The first step is to write out our equation: ax + b = c (where a, b, and c are integers). We want to find integer values of x that satisfy this equation. Now, if you’re like most people, your first instinct might be to start guessing at random numbers until you find one that works. But trust me when I say that there’s a better way!

Here’s the algorithm:
1. Divide both sides of the equation by gcd(a, b) (the greatest common divisor). This will give us an equivalent equation with smaller coefficients.
2. If c is now divisible by gcd(a, b), we have found a solution! Set x to be -b/gcd(a, b) and you’re done.
3. Otherwise, let d = (c mod gcd(a, b)) / a. This is the remainder of c divided by gcd(a, b), scaled up so that it has the same units as x.
4. If d is not an integer, then there are no solutions to this equation with integers for x and y.
5. Otherwise, let e be any integer such that (e * a) + (d * b) = gcd(a, b). This can always be done because we’re working in the ring of integers modulo gcd(a, b), which is a finite field with only finitely many elements.
6. Let x1 and y1 be any solutions to the equation (e * a) + (d * b)x = e * c mod gcd(a, b). These can always be found using the extended Euclidean algorithm or some other method for solving linear Diophantine equations in finite fields.
7. Let x2 and y2 be any solutions to the equation (e * a) + (d * b)y = -b mod gcd(a, b). These can also always be found using the extended Euclidean algorithm or some other method for solving linear Diophantine equations in finite fields.
8. Let x = x1 y2 and y = x2 mod a. This gives us two integer solutions to our original equation!
9. If you want more than one solution, just repeat steps 6-8 with different values of e (as long as they satisfy the condition in step 5).
10. And that’s it! You now have an efficient algorithm for solving linear diophantine equations. So go ahead and impress your friends with your newfound math skills!