This is what I read:
One can test if there exist integers x and y such that
$c\cdot x + j = d \cdot y + k$,
if $(k-j)\mod(\gcd(c,d)) = 0$
How can one see that?
PS: I understand what $\gcd $ and mod operations do.
PSS: Source: 1st frame of slide 6 within http://www.cs.arizona.edu/~collberg/Teaching/553/2011/Handouts/Handout-33.pdf