If d divides both k and one of a,b,c, then d divides

k - abc = 1.

If d is a positive integer, this implies d = 1.

Now apply that to the gcd's.

Assume k = dx, a = dy

then dx = dybc +1

1 = dx - dybc

1 = d (x - ybc), then d l 1, (x - ybc) integers

Good Luck.