A student erases a few of them, and instead writes the digit(s) of their product. (For example if he erases 4, 3 and 7 he would write the digits 8 and 4 since 4 * 3 * 7 = 84.) He also writes a few other random digits on the board.

He repeats this process until only one digit remains on the board. What is this digit and why?

The answer would be zero because of the 5.

If you multiply an odd number by 5 it'll end in 5 once all the odd numbers have gone you will have to multiply the 5 by an even number giving 0 which will then eventually give 0 at the end.

This could happen relatively quickly or take a very long time.