On Systems of Linear Diophantine Equations

Introduction Something happened to me recently I would wager has happened to many who read this note. Teaching a new topic, you cannot understand one of the proofs. Your first attempt to fill the gap fails. You look through your books for an answer. Next, you ask colleagues, go to the library, maybe even use the interlibrary loan. All in vain. Then it strikes you that, in fact, you cannot answer an even more basic and seemingly more interesting question. You peruse the books again. They seem to have answers to thousands of strange questions, but not to yours (the most natural one!). At the same time you cannot believe that your question could have been overlooked by generations of mathematicians. Days pass; the agony continues. Then one day, some way or other, you find the answer. In my case the answer was in a book I already owned. It followed from a theorem I had known for a long time, but I had never thought of this particular application. I must admit, indeed, that this theorem appeared in almost every book I had checked, but never with a pointer to this particular application, even as an exercise. Were the authors unaware of the application? Or did it seem to obvious to mention? In any case, here is the story.