smc

The "mathematical system" being alluded to is modulo $N$, so we shall be using such notation for convenience. From the problem, we know that $69\equiv 90\equiv 125\mathrm{mod}N$, so the difference between each of these numbers must be a multiple of $N$. $90-69=21=3\cdot 7$, and $125-90=35=5\cdot 7$. The only common factors between these two numbers are $1$ and $7$, but we know from the problem that $N\ne 1$. Thus, $N=7$. Now, $81\equiv 77+4\equiv 4\mathrm{mod}7$, so our answer is $\boxed{4}$.