Dear Hexin,

 Please consult

https://sites.math.rutgers.edu/~zeilberg/Hist21/mt1S.pdf

for the full solutions.

You got 72 points (out of 120).

Here is what you got wrong:

#1: -4 points  (P does not have to be a prime itself, but it must have a prime divisior different than p1,...,pk)

#3: -7 points (wrong way)

#4  (i): -2 points: logic is partially wrong

#8: - 5 points (see answer key)

#9: -10 points (not attempted)

#11: -10 points 

#12: -10 points  (see answer key, also conceptual error a0,a1, are NUMBERS not expressions in x)



 Best wishes,

   Dr. Z.