If $\Delta_{m,n}$ denotes the quotient $[u_m,v_n]/(u_m,v_n)$ of the lcm by the gcd, we obtain in this paper a lower bound for the greatest square-free factor $Q[\Delta_{m,n}]$ of $\Delta_{m,n}$ when $u_h=v_h, m>n$ (and $u_n\neq0$); this implies a lower bound for $\log Q[u_n]$ of the form $C(\log m)^2(\log\log m)^{-1}$, thereby improving on an earlier result of C. L. Stewart.