GCD of the numbers in a sequence (JBMO 2001)

For students of class 9-10 (age 14-16)
mutasimmim
Posts:107
Joined:Sun Dec 12, 2010 10:46 am
GCD of the numbers in a sequence (JBMO 2001)

Unread post by mutasimmim » Sun Sep 14, 2014 10:22 am

Let $A_n=2^{3n}+3^{6n+2}+5^{6n+2}$ for $n=0,1,2,....,1999$. Find the GCD of these numbers.

Nirjhor
Posts:136
Joined:Thu Aug 29, 2013 11:21 pm
Location:Varies.

Re: GCD of the numbers in a sequence (JBMO 2001)

Unread post by Nirjhor » Sun Sep 14, 2014 10:57 pm

Let the GCD be \(d\). \(A_0=35\) so \(d\in\{1,5,7,35\}\). \(A_1\equiv 4~\left(\bmod~10\right)\) so \(5\nmid d\). So \(d\in\{1,7\}\). Now \[A_n=8^n+9\cdot 27^{2n}+25\cdot 125^{2n}\equiv 1+9+25\equiv 0 \pmod{7}\] so \(d=\boxed{7}\)
- What is the value of the contour integral around Western Europe?

- Zero.

- Why?

- Because all the poles are in Eastern Europe.


Revive the IMO marathon.

mutasimmim
Posts:107
Joined:Sun Dec 12, 2010 10:46 am

Re: GCD of the numbers in a sequence (JBMO 2001)

Unread post by mutasimmim » Mon Sep 15, 2014 10:19 am

Similarly find out the GCD of the numbers of the form $n^{13}-n$ where $n$ is a natural number.

Post Reply