Let a,b,c be integers. Prove the following:

(a) if b|a, then b|ac

(b) if b|a and c|b, then c|a

(c) if c|a and c|b, then c|(ma + nb) for any integer m,n

Here is what I have come up with:

(a) I am unsure about how to prove.

(b) By definition, b|a iff a = qb

c|b iff b = qc

Therefore, a = qc => c|a

(c) By definition, c|a iff a = qc

c|b iff b = qc

Can anybody give me some pointers?