Divisor proof with absolute values

[reb659]

Homework Statement

I reduced a much harder problem to the following:
Prove that if abs(a-b) is divisible by k, and if abs(b-c) is divisible by k, then abs(a-c) is divisible by k.

Homework Equations

none really.

The Attempt at a Solution

I tried setting abs(a-b)/k = n and abs(b-c)/k = m where m and n are integers and trying to construct abs(a-c) from that but to no avail.
By the definition of absolute value, I know abs(x) = x when x>0 and =-x when x<0. I think trying all of the four possible cases might work, but would there be an easier way?

 

[Tedjn]

How do you get rid of the absolute values? If |x| is divisible by k, is x divisible by k?

 

[HallsofIvy]

(a- b)+ (b-c)= a- c.