b=ck for some integer k, 
a=bn for some integer n. 
substitute  into :
you get a = ckn
since kn is an integer:
you have a is divisble by c.
If a, b, c are different positive integers such that a is divisible by b,
and b is divisible by c, which of the following statements must be true?
I. a is divisible by c
II. a has at least 3 positive factors
III. a = bc
How can I prove without using numbers that only I and II are valid?
II depends on what you allow to be a factor. if a is allowed as a factor, then yes: a, b and c are obviously factors of a (by Goku's answer above).
if a factor must be "smaller than a", then no: consider a = 4, b = 2, c = 1.
to "disprove" III, it suffices to give a counter-example: let a = 12, b = 6, c = 3. then 12 is divisible by 6, and 6 is divisible by 3, but 12 is not 3*6.