6. First-year students admitted to the university were distributed into study groups so that each group had the same number of students. Due to a reduction in the number of specialties, the number of groups decreased by 9, and all first-year students were redistributed into groups; as a result, the groups were again equal in number, and each had fewer than 30 students. It is known that there were 2376 first-year students in total. How many groups are there now?

Answer: 99.
Let the new number of groups be $n$, then initially there were ( $n+9$ ) groups. In each group, there was an equal number of students, so $2376: n$ and $2376:(n+9)$. We factorize 2376 into prime factors ( $2376=2^{3} \cdot 3^{3} \cdot 11$ ) and list all divisors: $1,2,4,8,3,6,12,24,9,18,36,72,27,54$, $108,216,11,22,44,88,33,66,132,264,99,198,396,792,297,594,1188,2376$. According to the condition, the groups now contain fewer than 30 students, hence $\frac{2376}{n}<30$. This means we need to find among the divisors of 2376 two such that they differ by 9 and are greater than 79. These are 99 and 108. Therefore, $n=99$.