I'm stuck on the following problem, and I will appreciate it very much if anyone could direct me with to the right way of thinking, because I currently have no idea how to proceed after the initial attempt.
Question: If a store sells beers only in packages of $9$ cans and $14$ cans, prove that there is no way to buy exactly $103$ cans of beers from the store.
My Solution: First, since $9\nmid 103$ and $14\nmid 103$, there is no way to buy exactly 103 cans of beers with only 9-pack beers OR(exclusive) with only 14-pack beers.
However, if we mix and match the packs to get exactly $103$ cans of beers, eg) $x$ amount of $14$-packs plus $y$ amount of $9$-packs, I'm uncertain if the following method of proof is on the right track:
Second, since $(103 \mod 14) \equiv 5$ where the remainder is $5$ cans, and $(103 \mod 9)= 4$ has a remainder of $4$ cans. It's impossible to buy $5$ cans or $4$ cans of beers to add up to exactly $103$ cans. Therefore, it's impossible to get $103$ cans with the given packages of beers.