Prove that at least one integer in any set of ten consecutive integers is relatively prime to the others in the set.
Let the integers be mod 10:
0,1,2,3, ... 9 in any order
Clearly -out of 1,3,5,7,9 at most two are divisible by 3,only one by 5 and only one by 7, thus leaving
in the worst case one odd number that is not divisible by 2,3,5,7.
It does not have to be prime, but if it divides ,say, 11
its neighbors don't.
Edited on March 3, 2004, 8:42 am