King's Rare Jewels in Three Boxes

Let $X$ be the total number of jewels. $X/3 + kX/5 + 66 = X \implies 66 = X\left(1 - \frac{1}{3} - \frac{k}{5}\right) = X\left(\frac{2}{3} - \frac{k}{5}\right) = X\frac{10 - 3k}{15}$. So $X(10 - 3k) = 66 \times 15 = 990$. Since $k$ is a positive integer and $10 - 3k > 0$, possible values for $k$ are $1, 2, 3$. If $k = 1$: $10 - 3 = 7 \implies X = 990/7$ (not an integer). If $k = 2$: $10 - 6 = 4 \implies X = 990/4$ (not an integer). If $k = 3$: $10 - 9 = 1 \implies X = 990$. Thus $X = 990$ is uniquely determined.