For (c), start with the fact that we know an e value as in part (b) exists. Then, for example, consider m / e. If the remainder is not zero, try to produce something that contradicts part (b).

For (d), recall the fact that we've used before. If A | B and B | A then A = B. Try to use this for e and gcd(m,n).

Hope this helps a bit! This is one of the more abstract problems on the homework, and the proof is fairly difficult!