1. Demonstrate that there is a pair of integers such that the difference of them is divisible by 9. Solve by picking any 10 integers.

2. Looking at the first 2z integers 1, 2, . . . , 2z. Choose z + 1 of them. Then demonstrate that there is at least one pair of integers from the selection that are relatively prime. (A way to approach this problem is to show that at least two integers from the selection are consecutive.)

3. Show that when choosing 51 numbers selected from 1 to 100 will have a pair that its addition will result in 101