Possible scores 
A certain contest has three types of questions: part A, part B, and part C. There are 10 part A questions, 10 part B questions, and 5 part C questions. Each part A question is worth 5 points, each part B question is worth 6 points, and each part C question is worth 8 points. Not answering a question results in 2 points awarded for that question. Answering a question incorrectly results in 0 points for that question. How many scores from 0 to 150 are impossible to obtain?

I tried considering the possible "states" of the questions. Let $A_w$ represent the number of part $A$ questions answered wrongly, $A_u$ represent the number of part $A$ questions unanswered, and let $B_w, B_u, C_w,C_u$ be defined similarly. Let $1\leq x \leq 150.$ We want to consider the values of $x$ for which it is impossible to assign values to $A_w, A_u, B_w, B_u,C_w, C_u$ such that $0\leq A_w+A_u \leq 10, 0\leq B_w + B_u \leq 10, 0\leq C_w + C_u\leq 5$ so that $x-150 = -5A_w - 3A_u - 6B_w - 4B_u - 8C_w - 6C_u.$ We will list possibilities in the form $(C_w, C_u, B_w, B_u, A_w, A_u).$ The approach uses something similar to the greedy algorithm. Some cases are evidently impossible: 149, 148, 1. Others are definitely possible:


*

*$147 - (0,0,0,0,0,1)$

*$146 - (0,0,0,1,0,0)$

*$145 - (0,0,0,0,1,0)$

*$144 - (0,1,0,0,0,0)$

*$143 - (0,0,0,1,0,1)$

*$142 - (1,0,0,0,0,0)$

*$141 - (0,0,0,0,0,3)$

*$140 - (0,0,0,0,2,0)$
etc.


It seems that there are many scores that are obtainable, but I'm not sure how to show this efficiently. I think the answer is $4$- the only unobtainable ones are $1,3,149,148$, but I'm not sure how to show this.
Also, since there are $25$ questions in total and you get $2$ points for an unanswered question, any score of the form $2k, 1\leq k\leq 25$ is valid. Also, any score of the form $5k, 1\leq k\leq 10$ is valid, and similarly any score of the form $8k, 1\leq k\leq 5, 6k 1\leq k\leq 10$ is valid. This gives that $36$ scores from $1-60$ are obtainable.

Clarification: I made a typo in the original question. I should've said $10$ part $A$ questions and $5$ part $C$ questions instead of $5$ part $A$ questions and $10$ part $C$ questions.

 A: Disclaimer: This answer was based on an earlier version of the question.
I don't think this would necessarily be an easy problem with different parameters. But with these numbers, something stands out: you can easily get a solid range of scores just with the A questions. 
Namely, 


*

*with zero 5's, you can get any even number from 0 to 10;

*with one 5, you can get any odd number from 5 to 13;

*with two 5's, you can get any even number from 10 to 16;

*with three five's, you can get any odd number from 15 to 19;


In particular, you can get any score from 4 to 17 just with part A.
Now, if you start looking at what scores you can get by gradually adding correct B and (then) C questions, the gaps between them are small:


*

*0, 6, 12, ..., 54, 60, 68, 76, ..., 132, 140.


Since the increments are small, when you add to these a variable score on part A that can be anything from 4 to 17, there are no gaps left. Nothing is left out from 4 to 157.
Obviously, you can get a score of 2, but not 1 or 3. So the answer is that the only scores left out between 0 and 150 are 1 and 3.
