For every choice of non zero integers a,b,c, the integer lcm[a,c] divides

the integer lcm[a,b] * lcm[b,c] / b.

I was thinking of letting a=p/q, b=r/s, c=t/u. Then, somehow, the integer lcm[a,b] * lcm[b,c] / b will factor out an lcm[a,c], but I am not sure how to prove it that way.