Prove that b divides a if and only if (-b) divides a

Because it's if and only if, you need to prove in both directions.

1. prove that if b divides a then -b divides a.
suppose that b divides a. so a = bx (where x∈Z) by definition of "divides".
let x = -1 (-1 is an integer, so we can do this). so a = b(-1), so a = -b.
if a = -b then -b divides a (because any integer divides itself).

2. prove that if -b divides a then b divides a.
suppose that -b divides a. so a = -bx (where x∈Z) by definition of "divides".
a = -x(b) because multiplication is commutative.
let -x = k where k∈Z by closure under multiplication of integers.
a = kb. therefore b divides a by definition of "divides". 

thus we have proven that b divides a if and only if (-b) divides a.

i hope this is at least somewhat helpful-- different teachers like different methods/organization.