PLEASE ANSWER AND STATE ALL THEOREMS USED.

3. The Mathematics Department at Axis Mundi University has eight faculty members, each of whom is teaching three courses this semester. Their schedules are shown in the following table. Professor Courses Taught Abel 132, 136, 211 Banach 127, 131, 153 Cayley 131, 132, 211 Dynkin 127, 131, 205 Eilenberg 131, 138, 154 Frobenius 132, 136, 201 Gelfand 127, 131, 138 Hilbert 153, 154, 205 In arranging the final exams, it has been decided that each course will have a department final exam such that all sections of a given course will have their finals at the same time. Each professor must proctor his or her own exam. There is plenty of classroom space available, but everyone would like to get their exams finished as quickly as possible. What is the fewest number of time slots that can be used to give the exams? Your solution should include a table giving the exams scheduled and the professors proctoring during each time slot. 3. The Mathematics Department at Axis Mundi University has eight faculty members, each of whom is teaching three courses this semester. Their schedules are shown in the following table. Professor Courses Taught Abel 132, 136, 211 Banach 127, 131, 153 Cayley 131, 132, 211 Dynkin 127, 131, 205 Eilenberg 131, 138, 154 Frobenius 132, 136, 201 Gelfand 127, 131, 138 Hilbert 153, 154, 205 In arranging the final exams, it has been decided that each course will have a department final exam such that all sections of a given course will have their finals at the same time. Each professor must proctor his or her own exam. There is plenty of classroom space available, but everyone would like to get their exams finished as quickly as possible. What is the fewest number of time slots that can be used to give the exams? Your solution should include a table giving the exams scheduled and the professors proctoring during each time slot