For the proofs requested below, use facts and theorem given in this lesson, including results given
in the exercises, as justifications. Assume all letters represent integers.
(1) Mimic the proof given in the sample solutions for the proposition if a > 0 and b > 0, then
ab > 0 to prove:
(a) If a < 0 and b < 0, then ab > 0.
(b) If a < 0 and b > 0, then ab < 0.
(2) Prove that is m 2 = n 2 , then m = n or m = −n.
(Hints: (1) From algebra: a 2 − b 2 = (a + b)(a − b), and
(2) From exercises: If ab = 0, then either a = 0 or b = 0.)
(3) Determine all the integers that 0 divides.
Hint: Think carefully about the definition of the divides relation! This question is
about the divides relation, not about the arithmetic operation of division (a maybe subtle
distinction). The correct answer is probably not what you might first think it is.
(4) Prove: For integers r,s,t,and u, if r|t and s|u, then rs|tu.
(5) Determine the quotient and remainder when 117653 is divided by 27869. (Finally, an easy
(6) (bonus) Prove or give a counterexample: If p is a prime, then 6p + 1 is a prime.