have to do a couple problems for homework and im not understanding exactly what is going on and why its the answer

27. Let m, n, and d be integers. Show that if d \ m and d \ n, then d\ (m - n).

my professor gave us the answer to this but i do not understand why it works

answer: m = dq1 n = dq2 m - n = dq1 - dq2 = d(q1 - q2) therefore d \ (m - n)

these are the other problems i have to do

28. Let m, n, and d be integers. Show that if d \ m, then d \ mn.

31. Let a, b, and c be integers. Show that if a \ b and b \ c, then a \ c.

33. Give an example of consecutive primes p1 = 2, p2 ..., pn

where

p1p2... pn + 1

is not prime.

any help would be much appreciated