For each integer $N>1$, there is a mathematical system in which two or more positive integers are defined to be congruent if they leave the same non-negative remainder when divided by $N$. If $69,90,$ and $125$ are congruent in one such system, then in that same system, $81$ is congruent to

$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad \textbf{(E) }8$
1. We are given that the integers \(69\), \(90\), and \(125\) are congruent modulo \(N\). This means:
   \[
   69 \equiv 90 \equiv 125 \pmod{N}
   \]
   For these numbers to be congruent modulo \(N\), the differences between any pair of these numbers must be divisible by \(N\).

2. Calculate the differences:
   \[
   90 - 69 = 21
   \]
   \[
   125 - 90 = 35
   \]
   \[
   125 - 69 = 56
   \]

3. The value of \(N\) must be a common divisor of \(21\), \(35\), and \(56\). Find the greatest common divisor (GCD) of these numbers:
   \[
   \text{GCD}(21, 35, 56)
   \]

4. Factorize each number:
   \[
   21 = 3 \times 7
   \]
   \[
   35 = 5 \times 7
   \]
   \[
   56 = 2^3 \times 7
   \]

5. The common factor is \(7\). Therefore, \(N = 7\).

6. Now, we need to determine the congruence of \(81\) modulo \(7\):
   \[
   81 \div 7 = 11 \text{ remainder } 4
   \]
   Thus,
   \[
   81 \equiv 4 \pmod{7}
   \]

Conclusion:
\[
\boxed{4}
\]