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. Walters entered?
$\textbf{(A)} \ 71 \qquad \textbf{(B)} \ 76 \qquad \textbf{(C)} \ 80 \qquad \textbf{(D)} \ 82 \qquad \textbf{(E)} \ 91$
Solution 1
The first number is divisible by $1$.
The sum of the first two numbers is even.
The sum of the first three numbers is divisible by $3.$
The sum of the first four numbers is divisible by $4.$
The sum of the first five numbers is $400.$
Since $400$ is divisible by $4,$ the last score must also be divisible by $4.$ Therefore, the last score is either $76$ or $80.$
Case 1: $76$ is the last number entered.
Since $400\equiv 76\equiv 1\pmod{3}$, the fourth number must be divisible by $3,$ but none of the scores are divisible by $3.$
Case 2: $80$ is the last number entered.
Since $80\equiv 2\pmod{3}$, the fourth number must be $2\pmod{3}$. The only number which satisfies this is $71$. The next number must be $91$ since the sum of the first two numbers is even. So the only arrangement of the scores $76, 82, 91, 71, 80$ or $82, 76, 91, 71, 80$ $\Rightarrow \text{(C)}$

Solution 2
We know the first sum of the first three numbers must be divisible by $3,$ so we write out all $5$ numbers $\pmod{3}$, which gives $2,1,2,1,1,$ respectively. Clearly, the only way to get a number divisible by $3$ by adding three of these is by adding the three ones. So those must go first. Now we have an odd sum, and since the next average must be divisible by $4, 71$ must be next. That leaves $80$ for last, so the answer is $\mathrm{C}$.

Solution 3
we know that the average of the scores is an integer 
so that means s1+s2+s3+s4 must be an even number divisible by 4 
we have 3 even scores and 2 odd scores
which means that the last score cannot be odd because otherwise, we would get an odd number divided by an even number in the denominator.
so we have answers that are even 
76,80,82
We see 3 cases 
where 76 is the last score, 80 is the last score, and 82 is the last score  
76= 1 mod(5)
which means 
80+82+71+91= 0 mod(4)
80+82+71+91= 4 mod(5)
80= 0 mod(5)
76+82+71+91= 0 mod(5)
76+82+71+91= 0 mod(4)
82= 2 mod(5)
76+80+71+91= 0 mod(4)
76+80+71+91= 3 mod(5)
Case 1: 76
324 is divisible by 4 
324 divided by 5 is 1
which means 76 is not the last number
case 2:
324-4=320
320 is divisible by 4 
320 is divisible by 5 
which means this case is true. 
case 3:
320-2=318
318 is not divisible by 4 which makes it incorrect even though 
it has a remainder of 3 when divided by 5