1957 AHSME Problem 24

If the square of a number of two digits is decreased by the square of the number formed by reversing the digits, then the result is not always divisible by:

\textbf{(A)}\ 9 \qquad \textbf{(B)}\ \text{the product of the digits}\qquad \textbf{(C)}\ \text{the sum of the digits}\qquad \textbf{(D)}\ \text{the difference of the digits}\qquad \textbf{(E)}\ 11

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.