Problem: Show that for n, a natural number, 5 divides 6^(n)-1 is true.
Conversation between Captain Kirk, Spock, McCoy, Scotty and
Uhura:
Location:
Enterprise bridge.
Uhura: Captain,
we are receiving a message from the Planet Modulo.
Captain Kirk: View screen on.
Rep from Modulo: Your ship is passing into our territory. We
need to know your mission.
Uhura mutes the view screen.
Capt Kirk: Spock, what should we say?
Spock
and Kirk whisper.
Capt Kirk: View screen on. We are here to
show that the divide between our people is not that great. For all of the
natural people of Modulo, we will show you that our modular 5 is the only thing
divides the Six nations natural powers, and that the separation is only a difference
of 1.
Rep from Modulo: We welcome you and look forward to you
showing us the way.
Kirk:
View screen off.
Kirk:
Scotty, do we have any mod 5?
Scotty:
Aye Captain! Here is one.
Scotty
hands Kirk a mod 5.
McCoy: But they are
dead Jim…..
Ignoring McCoy, Kirk: Let’s get the mod 5 working. Spock, how
about the natural powers of the Six nations.
Spock: Here Captain. Six to the natural powers. I adjusted it so there is a difference of
one.
Kirk: Now we need to
show that it is true. We can induce the truth out of it. But first we need to
step into the Basis. Scotty what do we have?
Scotty: We have that
if the natural power is one then 5 equals 5!
McCoy: But they are
deader than dead Jim.
Ignoring McCoy, Kirk: Scotty,
do we have the k and the k+1? We need it
now!
Scotty: But that’s
not possible!! It will take hours and hours!
Kirk: Now!
Scotty: We have them!
Kirk: Spock, set up
the mod 5!
Spock: We know the
truth that 5 divides the Six nations to the k power with a difference of 1.
According to the Laws of Modulo, we also know that Six to the k powers is
congruent to 1 times the mod 5! We also know that as this is true for k, the
Laws of Modulo state that this is also true for the powers of k+1.
Kirk: Hurry, Scotty.
Scotty: I’m working as fast as I can. I can’t find the divide!
Spock: Try using the Division Algorithm.
Scotty: That did it! 5
now divides the Six nations to the powers of k+1 by only a difference of 1!
Kirk: Hand it to me!
View screen on. We, from the Starship Enterprise, would like to give the Planet
Modulo the truth that we have found. Our
Mod 5 has been used to show that 5 is the only thing divides the Six nations
natural powers, and that the separation is only a difference of 1.
Rep from Modulo:
Thank you. We welcome you to our Planet.
The crew of the Starship
Enterprise transports to Modulo’s surface.
No one ever found out to whom McCoy was referring………
