Is there an easier way to prove the divisibility of integers?

[tronter]

Is there an easier way...

to prove http://img410.imageshack.us/my.php?image=linug1.jpg"?

I did a proof by cases.

 

[cellotim]

Well, consider the sets of integers (1,...,n) (n+1,...,2n) and so on (and the negatives likewise). Now, these partition the integers. Take n consecutive integers anywhere in the integers and for none of them to divide n exactly you would have to fit them inside one of these sets. That's impossible of course. For two to divide n, you would have to overlap mn and (m+1)n for some integer m. That's also impossible without at least n+1 integers. QED.