Can someone help me prove or disprove this statement:

For all integers a,b, and c, if a divides b (and results in some integer) and a cannot divide c (and result in some integer) then a cannot divide b+c (and result in some integer).

I have done a lot of work on this and im having a hard time, I did not find out much except that :

if a|b and b|c then a|c by transitivity

also, if a|b and b|c then a|b+c.

Thank you:)