A) Admirable solution, nicely presented.
B) Working backwards is a good problem solving strategy, but make sure to present solutions arguing forward from known information and facts.
C) Complete solution, or close enough; fine job.
D) The proof was difficult to decipher because of a confusing or illegible presentation.
E) The paper did not merit any points, but the response was quite enjoyable to read!
F) There was a mistake performing congruence arithmetic. For example, if m is even, then 2r=–2 mod m leads to r=–1 mod (m/2), not mod m. Similarly, 2(r+1)=0 mod m is not equivalent to 2=0 mod m or r+1=0 mod m.
H) Half the question was answered correctly; the other half was omitted or no headway was made.
J) The proof was hard to understand due to confusing or faulty logical structure.
K) Unfortunately, the factorizations listed are actually equivalent, given the value of m chosen. For instance, (x+6)(x-4) and (x+9)(x-7) are the same factorizations mod 3.
L) Fine answer, but more work than necessary; it is OK to be more concise or cite previous results.

M) Mostly there; main ideas are correct but points deducted for missing details or too brief a proof.
N) Not bad; careless mistakes or a false statement tarnish an otherwise correct solution.
O) Omitted problem or no attempt at a proof.
P) Please include an explanation of why the primes you give do not lead to factorizations. (As done for mod 7 in part i.)
R) On the right track or a few of the correct ideas present, so deserving of some credit.
S) You overlooked some entries in your lists of primes OR you forgot to make a conjecture based on your results.
T) Note that (x-5)(x+7) is the same factorization as (x-12)(x+14) when working mod 19.
U) Your mathematics and/or presentation was a little sloppy!
W) On the wrong track or a very difficult approach, but warranting some credit.
X) The proof did not check the case m=4 (which must be considered separately) due to an earlier mistake involving congruence arithmetic.
Y) Little or no significant progress towards a solution (occasionally despite a fair amount of work), or misinterpretion of the question.