EE-MO 2007

Find the largest integer such that every number after the first is one less than the previous one and is divisible by each of its own numbers.

The headteacher wants to hire a certain number of new teachers in addition to existing teachers. If he hired an additional $10$ teachers, the number of school students would be reduced number per teacher by $5$ . However, if the headmaster hired $20$ new teachers, the number of students per teacher would be reduced by $8$ . How many students and how many there are teachers in this school?