Diophantine Equations

A Diophantine equation is an equation where you’re only allowed whole number (integers!) solutions, and the variables (like x and y) represent counts of things (like apples, tickets, or llamas). The simplest Diophantine equation is like:

ax+by=c 

You're trying to find integers x and y that make the equation true.  The fact that such equations often have more than one answer often make beginner uncomfortable, since they are used to the "one-variable, one-equation, one-solution" routine with certainty. Many times, questions involving Diophantine equations require more than algebra to solve -- it might involve counting (how many solutions are there?), number theory (for example, the use of remainders), and many more tricks.  


1987 AJHSME Problem 24


Ask any mathlete about Simon’s Favorite Factoring Trick, and they can probably tell you a story about how it saved them in a tricky problem. It's one of those clever little tools that feels like magic once you see it in action — especially when a seemingly messy expression suddenly turns into a clean, factorable form.

At its core, Simon's trick is all about making the expression factorable by completing the rectangle (a cousin of completing the square). When you see something like

𝑥𝑦 + 𝑎𝑥 + 𝑏𝑦 + 𝑐

you add and subtract the right constant to group it neatly — turning chaos into a product of binomials. It’s a small move with a big payoff, and once you master it, you’ll start spotting its potential in unexpected places — like Diophantine equations.