jmc

Let $n$ be the number of dimes Natasha has. We know that $10<n<100$. The stacking data can be rephrased as $$\begin{array}{rl}n& \equiv 1(\mathrm{mod}3)\\ n& \equiv 1(\mathrm{mod}4)\\ n& \equiv 1(\mathrm{mod}5).\end{array}$$ Notice that any number $n$ such that $n\equiv 1(\mathrm{mod}60)$ solves this system. (The Chinese Remainder Theorem tells us that 1 is the only residue class modulo 60 that solves all of these equivalences.) Therefore $n=\boxed{61}$ is between 10 and 100 and solves this system.