GCD of the numbers in a sequence (JBMO 2001)
Posted: 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.
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 \(d=\boxed{7}\)