April POM: Kimball’s Cymbals

Kimball operates a thriving music store to support Nompton’s local artists.  All the equipment is handmade and cutting edge, but the best bargain is the drums and cymbals.  You should buy the cymbals in bulk, because Kimball only sells them in sets of 5 and 7.  Kimball insists that for really large orders, you can figure out a way to break it up into the 5-cymbal sets and the 7-cymbal sets.  Often in multiple ways.  For instance, if you’re buying 70 cymbals, you can simply buy 10 of the 7-sets.  You can also buy 7 of the 5-sets, together with 5 of the 7-sets.

• What is the largest number of cymbals that you can’t buy as a combination of these sets?
• Suppose instead that the cymbals come in sets of 4 and 7.  What is the largest number that you can’t buy?
• Same question for sets of 9 and 13.
•  Suppose that the cymbals come in packages of $a$ and $b$, for positive integers $a$ and $b$.  When are there infinitely many numbers that you can’t buy?    In the other case, what is the largest number that you can’t buy?

To win, you need to answer the first three problems, showing your work.  After that, we will look for a winner with a correct answer for the fourth question, with the best explanations.  See the rules for more information.