If then for some . Clearly then and since the conclusion follows.

Sorry, I just noticed this. What it means to say that is that divides or that . Does that make a little more sense? So if thenmy 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)

so then or and since the integers are closed under multiplication the conclusion follows.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.

These all can be done the same. and . So then and the conclusion follows.31. Let a, b, and c be integers. Show that if a \ b and b \ c, then a \ c.

What does "consecutive" mean? That or that is merely the next prime in the list of primes? I'm assuming the latter. What do you think?

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

where

p1p2... pn + 1

is not prime.