Proofs Via the Contrapositive

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

This discussion gives you practice proving a statement by proving its contrapositive.

Note that this discussion has something for you to turn in! In particular, come to your next grading meeting with me prepared to summarize the proof developed in this discussion in your own words (bringing notes from the discussion is OK). I will lightly grade that summary, and count the grade toward the “proof using the contrapositive” learning outcome (outcome 5.2).

Prove the following:

Conjecture: For every integer n, if an integer x is not divisible by n, then x is also not divisible by every integer multiple of n.

I strongly encourage you to work collectively on this, i.e., I don’t expect any one person to create the entire proof. Rather, I expect you to share ideas, suggestions, questions, etc. through this discussion until someone, or maybe a few someones, can put some of them together into a complete proof. What each of you will then bring to your next grading meeting will be your paraphrase or summary of the final proof, as you understand it. This can, and I expect generally will be, your understanding of work that other people made big contributions to, I don’t require it to be entirely your own proof.