7 divide $n^7-n$

For students of class 9-10 (age 14-16)
tarek like math
Posts:56
Joined:Fri Feb 18, 2011 11:30 pm
7 divide $n^7-n$

Unread post by tarek like math » Sat May 07, 2011 12:50 am

Prove $n^7-n$ is divisible by 7.
it's actually true for some $n^p-n$ but not proven for all p=prime.
Hint: multiplication of 7consecutive numbers divisible by 7.

User avatar
Tahmid Hasan
Posts:665
Joined:Thu Dec 09, 2010 5:34 pm
Location:Khulna,Bangladesh.

Re: 7 divide $n^7-n$

Unread post by Tahmid Hasan » Sat May 07, 2011 11:28 am

induction works too.
বড় ভালবাসি তোমায়,মা

User avatar
Nadim Ul Abrar
Posts:244
Joined:Sat May 07, 2011 12:36 pm
Location:B.A.R.D , kotbari , Comilla

Re: 7 divide $n^7-n$

Unread post by Nadim Ul Abrar » Sat May 07, 2011 12:48 pm

n^6=1 mod 7
=> n^7/n= 1 mod 7
=> n^7-n=0 mod 7

tarek like math
Posts:56
Joined:Fri Feb 18, 2011 11:30 pm

Re: 7 divide $n^7-n$

Unread post by tarek like math » Sat May 07, 2011 11:13 pm

do u please can explain it ? which theorem u use?
do anyone can solve it without induction?

tushar7
Posts:101
Joined:Tue Dec 07, 2010 3:23 pm

Re: 7 divide $n^7-n$

Unread post by tushar7 » Wed May 11, 2011 3:14 pm

nadim used fermat's little theorem

tarek like math
Posts:56
Joined:Fri Feb 18, 2011 11:30 pm

Re: 7 divide $n^7-n$

Unread post by tarek like math » Wed May 11, 2011 5:22 pm

Does he can prove it for all $n^p-n$ ?

User avatar
Masum
Posts:592
Joined:Tue Dec 07, 2010 1:12 pm
Location:Dhaka,Bangladesh

Re: 7 divide $n^7-n$

Unread post by Masum » Thu May 12, 2011 2:28 pm

The fact is whether we can use fermat's theorem or not.
One one thing is neutral in the universe, that is $0$.

Post Reply