[testplzignore]

Paradoxically, this may be the best answer because it is not the best answer.

In a room of 8500 people, it is likely that there is someone in the room smarter than you and knows a better way to pick random numbers. Come into the room and tell everyone that your objective is to pick a random number from 1-10. Maybe even go as far as to tell them your mod 10 idea. Wait a bit to let them to think about it, then start asking for the random numbers. It is likely that someone will have come up with a better way to pick a random number than your mod 10 solution, and thus their answer will be more uniformly random that what your solution can produce. And thus your solution becomes at least as uniformly random as theirs.