if a divides b but a does not divide c, then a does not?

Question

(Prove: For all integers a, b, c, if a divides b but a does not divide c, then a does not divide b+c.)

Answer 1

Since a|b, we have b = az for some integer z.
Since a\c, we have c = ay+r for some r > 0 and < a.

(I am using \ to mean "does not divide".)

Therefore

b+c = az+ay+r = a(z+y)+r

so that when you divide b+c by a you get a remainder r which is not equal to zero and is less than a. Therefore, a\(b+c).

Answer 2

if a divides b, then b is a multiple of a...lets call this multiple k
b = ka

b + c = ka + c
dividing ka + c by a = k +c/a, but since c does not divide by a, it is false.