Admission to Exam

Last semester, students from the Faculty of Mechanics and Mathematics at the fictional university had to take an exam in network technologies. $N$ professors teaching the subject agreed on the following arrangement: during the semester, there would be N^2 lab assignments. The first professor would conduct labs numbered $1$, N+1, 2N+1, ..., N^2-N+1; the second professor — labs numbered $2$, N+2, 2N+2, ..., N^2-N+2, and so on. The $N$-th professor would conduct labs numbered $N$, $2N$, $3N$, ..., N^2. The professors also recalled that in recent years, lazy students had been skipping many lab assignments, leading to poor exam scores. Therefore, they decided that a student would be allowed to take the exam only if they've attended at least one lab for each professor.

$N$ students, living in the same dorm room, did not know how many labs would be held during the semester or how many professors taught them. Each student had a different attitude towards study: the first student attended all labs throughout the semester, the second attended only labs with numbers divisible by two, the third only those divisible by three, and so on. After all lab assignments were completed, it was found that only $K$ of these students were allowed to take the exam.