Question: Our goal in this problem is to determine when the converse of Theorem holds and when it does not, namely, when does ac ≡ bc (mod n) imply that a ≡ b (mod n)?
(a) Let us recall our counterexample: 18 ≡ 24 (mod 6), but 9?12 (mod 6). In fact, 18 ≡ 24 ≡ 0 (mod 6). Find another example in which ac ≡ bc ≡ 0 (mod n) and a ? b (mod n). (Try not to have n = 6.)
(b) In your example, was n even? If so, find another example in which n is odd.
(c) Make a conjecture: under what conditions does the converse of Theorem hold?
(d) Challenge: Perhaps there is something special about zero . . . or perhaps not. Use the definition of congruence modulo n to figure out whether there are a,b,c,n such that ac ≡ bc (mod n) and ac ? 0 (mod n) and a ? b (mod n).
Theorem: Let a,b,c∈ Z and n∈ N. If a ≡ b (mod n), then ac ≡ bc (mod n).