1. Let n be any positive integer. Prove or disprove: any set of ten consecutive positive integers {n, n + 1, . . . , n + 9} contains at least one prime number. 2. (Thanks to the NPR radio show Car...

1. Let n be any positive integer. Prove or disprove: any set of ten consecutive positive integers {n, n + 1, . . . , n + 9} contains at least one prime number.

2. (Thanks to the NPR radio show Car Talk, from which I learned this exercise.) Imagine a junior high school, with 100 lockers, numbered 1 through 100. All lockers are initially closed. There are 100 students, each of whom—brimming with teenage angst—systematically goes through the lockers and slams some of them shut and yanks some of them open. Specifically, in round i := 1, 2, . . . , 100, student #i changes the state of every ith locker: if the door is open, then it’s slammed shut; if the door is closed, then it’s opened. (So student #1 opens them all, student #2 closes all the even-numbered lockers, etc.) Which lockers are open after this whole process is over? Prove your answer.