jmc

Since the geometric sequence is strictly decreasing, the common ratio $m/7$ must be a positive number between 0 or 1. For, if it were greater than 1, the sequence would keep increasing, since the first term is positive. If the ratio was 0, then the sequence would consist of 0's after the first term and would not be strictly decreasing. Finally, if the ratio was negative, the sequence would alternate between positive and negative terms, and thus would not be decreasing. So we have $0<\frac{m}{7}<1$, or $0<m<7$. There are $\boxed{6}$ possible integer values of $m$: 1 , 2, 3, 4, 5, 6.