jmc

Recall that, by definition, $531n\equiv 1067n(\mathrm{mod}24)$ means that $531n-1067n$ is divisible by 24. In other words, $$\frac{1067n-531n}{24}=\frac{536n}{24}=\frac{67n}{3}$$must be an integer. Since $67$ and $3$ are relatively prime, $n$ must be a multiple of $3$, the smallest of which is $\boxed{3}$.