You know that there exists two integers a and x such that:ax=k

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

thus:x=k/a

then:x=k/(nc)

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.