1987 AJHSME Problem 19

A calculator has a squaring key \underline{\,x^2\,} which replaces the current number displayed with its square. For example, if the display is \underline{\,000003\,} and the \underline{\,x^2\,} key is depressed, then the display becomes \underline{\,000009\,}. If the display reads \underline{\,000002\,}, how many times must you depress the \underline{\,x^2\,} key to produce a displayed number greater than 500?

\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 250

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