Consider six consecutive positive integers. Show that there is a prime number that divides exactly one of them.

Let p be the greatest prime number that divides at least one of the six consecutive numbers.

p can't be 2 or 3 because among the six numbers at least one divides by 5.

If p>=7 then the number that divides by p is the only one, because is a multiple of p and among the six numbers there is only one that has this property.

So, it remains the problem when p=5. I think that the only case is for 1, 2, 3, 4, 5, 6. Among any other six consecutive numbers there is at least one that divides by a prime greater than 5. But this must be prooved.

Posted by Stefan on 2006-08-30 04:08:06