Division rule

2. Division Rule (14 points) (1) (2 points) How many positive integers not exceeding 100 are divisible either by 6 or by 9? (Hint: LCM (6,9) 18) (2) (6 points) Consider the following seating arrangment problems: (a) How many ways can five out of twelve people be seated around a circular table? (b) How many ways can five out of twelve people be seated around a circular table where two seatings are considered the same when ev- eryone has the same immediate left and right neighbors? (c) How many ways can five out of twelve people be seated arounda circular table where two seatings are considered the same when ev- eryone has the same two neighbors (i.e., without regard to whether they are left or right neighbors)? (3) (6 points) Suppose that p and q are prime numbers and that n -pq. Use the principle of inclusion-exclusion to find the number of positive integers not exceeding n that are relatively prime to n