A large University employs professors, lecturers and security guards. Each one of the ten thousand employees receives exactly the same salary, paid in cash on the first of the month. On every day of the month, one of the professors takes part of his or her cash and divides it equally among all the lecturers. Then a lecturer distributes part of his or her money equally among all the professors. No one gives cash to security guards or uses it in any other way.
At the end of the month, Professor Alice and Lecturer Bob were left with a thousand pounds in cash each, whereas Professor Chuck had only a pound and a penny, the same as Lecturer Dave. If there are more lecturers than professors, how many security guards work at the University?