Consider six consecutive positive integers. Show that there is a prime number that divides exactly one of them.
(In reply to
Proof by Tristan)
Absolutely no problem with Tristan's proof, but here is a simpler version, based on his idea:
No
matter what set is chosen, exactly three of the integers are
even. Of the remaining three (which differ from each other by
either 2 or 4), at most one is divisible by 3 and at most one is
divisible by 5. So at least one is not divisible by 2, 3, or
5. Consider the first number in the set not divisible by 2, 3, or
5.
If it is the digit 1, then the set is necessarily 123456, and 5 divides only one number in the set.
If it is not 1, then it is necessarily divisible by a prime
greater than 5, and that prime cannot divide any other numbers in the
set.
QED.
A SUSPICION:
There is nothing magic about the number 6. I suspect that in any
set of n consecutive positive integers there exists at least one prime
number that divides exactly one of them (unless the set contains only
the digit 1).
re: Proof Simplified, and a suspicion
