Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walter entered.
The sum of first three numbers must be a multiple of $3$. We note that the residues modulo $3$ of these five numbers are $2$, $1$, $2$, $1$, and $1$, respectively. Therefore $76$, $82$, and $91$ must be entered first.
The sum of these three numbers is odd. In order for the sum of the first four numbers is a multiple of $4$, the fourth number must be odd which has to be $71$. Hence the last score entered must be $\boxed{80}$.