How many integer quadruples $a,b,c,d$ are there such that $7$ divides $ab-cd$ where $0\leq a,b,c,d < 7$?

$ \textbf{(A)}\ 412
\qquad\textbf{(B)}\ 385 
\qquad\textbf{(C)}\ 294
\qquad\textbf{(D)}\ 252
\qquad\textbf{(E)}\ \text{None}
$
1. We need to find the number of integer quadruples \((a, b, c, d)\) such that \(7\) divides \(ab - cd\). This is equivalent to finding the number of solutions to the congruence \(ab \equiv cd \pmod{7}\).

2. First, we note that \(a, b, c, d\) can each take any integer value from \(0\) to \(6\). Therefore, there are \(7\) possible values for each variable.

3. We will use the multiplication table modulo \(7\) to determine the number of distinct products \(ab \pmod{7}\) and \(cd \pmod{7}\).

4. The multiplication table modulo \(7\) is as follows:
   \[
   \begin{array}{c|ccccccc}
     \times & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
     \hline
     0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
     1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
     2 & 0 & 2 & 4 & 6 & 1 & 3 & 5 \\
     3 & 0 & 3 & 6 & 2 & 5 & 1 & 4 \\
     4 & 0 & 4 & 1 & 5 & 2 & 6 & 3 \\
     5 & 0 & 5 & 3 & 1 & 6 & 4 & 2 \\
     6 & 0 & 6 & 5 & 4 & 3 & 2 & 1 \\
   \end{array}
   \]

5. From the table, we observe that:
   - There are \(13\) pairs \((a, b)\) such that \(ab \equiv 0 \pmod{7}\).
   - For each non-zero residue \(1, 2, 3, 4, 5, 6\), there are \(6\) pairs \((a, b)\) such that \(ab \equiv k \pmod{7}\).

6. We need to count the number of ways to pick two pairs \((a, b)\) and \((c, d)\) such that \(ab \equiv cd \pmod{7}\). We consider each residue class separately:
   - For \(ab \equiv 0 \pmod{7}\), there are \(13\) choices for \((a, b)\) and \(13\) choices for \((c, d)\). Thus, there are \(13^2 = 169\) quadruples.
   - For each non-zero residue \(k\), there are \(6\) choices for \((a, b)\) and \(6\) choices for \((c, d)\). Thus, for each \(k\), there are \(6^2 = 36\) quadruples. Since there are \(6\) non-zero residues, the total number of quadruples for non-zero residues is \(6 \cdot 36 = 216\).

7. Adding these together, the total number of quadruples \((a, b, c, d)\) such that \(ab \equiv cd \pmod{7}\) is:
   \[
   169 + 216 = 385
   \]

The final answer is \(\boxed{385}\)