Number Theory: Problem 2

You thought you'd find the model solution here, didn't you? Maybe you are ready for it, maybe not. If you are stuck, look at the hints below. If you think you have a working proof, first do the elementary checks below. There's no point in looking at the model solution until you have something that might be close to right.

Hints

The hypothesis of the claim gives you facts about divisibility. The desired conclusion also uses the divides operator. Did you use the definition of divides to convert these divisibility equations into equations that just use plain algebra without the divides operator? If you do this to the hypothesis and conclusions, you can convert our original problem into a familiar algebra problem.

You are reading the divides operator in the correct order, right? That is, the small number is on the left and the big number is on the right.

Also, you could look at the example proofs in the textbook. Some of them are very similar to what you'll be writing here. Maybe you can use one of them as a model and adapt it?

Self-check

Did you introduce the variables p, q, and r at the start of your proof? Did you specify that they are all integers? The definitions of these variables in the the claim are local to the claim, so these variables are no longer defined when you start the proof.

Right after you introduce the variables, you should state the given information, i. the hypothesis of the if/then statement in the claim. If the hypothesis in the claim is P, you should write "suppose that P" at the start of your proof.

Are you sure that you're using our official definition of divides? If you don't have it memorized, did you look it up? The instructions said not to use facts about divisibility. That means don't use lemmas such as "if p|q and p|r, then p|(p+r)."

Do you have a fraction or a division sign anywhere in your proof? You shouldn't.

Does your proof end at the conclusion of the if/then statement in the claim? You'll lose points if you stop one step earlier and hope the reader will then look back at the claim for the conclusion.

If you have any of these issues, fix your written-out solution before moving forwards to the full annotated solution.