Let $N$ be a positive integer. It is true that $1|N, 2|N, 3|N, \dots, 31|N$, except two. Which are false?
It seems like the problem is impossible. For example, any two consecutive integers must have an even integer E, and this even integer is divisible by 2 and an integer less than it, and since 2 and that integer divide N, E must divide $N$, which is a contradiction. For example, if we chose 31 and 30, then since 2 and 15 divide N, it must be the case that 30 divides $N$, which is a contradiction. So I'm not sure where to go with this problem.