You know that there exists two intergers b and y such that: by=k
Let us say that the least common multiple of a and b is n
Thus there exists two integers n and c such that: nc=a
Thus there exists two integers n and d such that: nd=b
Go back to: ax=k
multiply both sides by c: xc=k/n
Now, two integers multiplied together equal an integer, so k/n is an integer, thus k is divisible by n.