Extensions of Chicken Nugget Theorem

Question

Given that $a,b,c\ $ are pairwise relatively prime, what is the largest number not expressible as a linear combination of $a,b,c\ $?

What if $a$ and $b$ have common factor $m$?

What if $a$ and $b$ have common factor $m$, and $b$ and $c$ have common factor $n$?

Answer 1

This is more of an extended comment than an answer to your specific questions.

This is called the Frobenius number of the semigroup generated by $a,b$ and $c$. The case for two generators, say $a$ and $b$ with $a$ and $b$ rel. prime, has a closed formula given by $$ g(a,b)=ab-a-b.$$

However, in general for 3 generators it is known that there is no polynomial formula expressing the Frobenius number in terms of $a,b$ and $c$. The reference is F. Curtis (1990). "On formulas for the Frobenius number of a numerical semigroup". Mathematica Scandinavica. 67 (2): 190–192. You can find the paper here: https://www.mscand.dk/article/view/12330/10346

You can read more about this in: https://en.wikipedia.org/wiki/Coin_problem#Statement or https://en.wikipedia.org/wiki/Numerical_semigroup

Answer 2

Your not all relatively prime cases, are restricted cases of the 2 number cases: $$a=md\land b=me \implies m(dx+ey)+cz=f$$

aka what are the values that can't be made by combinations of d and e, multiply them by m and ask what numbers they can't reach by adding an integer multiple of c to them.

Your all relatively prime case, I'm not sure about except by looking at the other answer.