Assuming that we are dealing with the integers.
Now, I ask you: if a divides b then does a divide (bc)?
If that is your proof then it isn't correct. You are to assume a|b or a|c, not necessarily both.
I am going to assume that a divides only b. (We can simply switch the roles of b and c if a divides c and not b, so we are losing no generality here by doing this.)
So a|b implies that b = na, where n is some positive integer. Thus
bc = (na)c = (nc)a
But nc is also a positive integer. Thus a|bc as well.
-Dan