<s>[INST] Find GCD$(A_0,A_1,...,A_{2015})$ where $A_n=2^{3n}+3^{6n+2}+5^{6n+2}$ ; $n=0,...,2015$ Find GCD$(A_0,A_1,...,A_{2015})$ where $A_n=2^{3n}+3^{6n+2}+5^{6n+2}$ ; $n=0,...,2015$
Is there some intuitive method or formula for finding GCD of $n$ integers?
 [/INST] $n=0$ gives $1+9+25 = 35$, so it's either $1, 5, 7$ or $35$. 
$n=1$ gives $8+ 3^8 + 5^8 = 397194 = 2 \cdot 3 \cdot 7^3 \cdot 193$ so it could be $1$ or $7$.
Let's see what happens mod  $7$ (this means we look at all numbers ignoring powers of $7$, so basically any number becomes $0,1,2,3,4,5$ or $6$ according to the rest of its division by $7$)
$2^{3n} = 8^n = 1^n = 1$ 
$3^{6n+2} = (9)(3^{6n}) = 2(2^{3n}) = 2(1)^n = 2$ 
$5^{6n+2} = (25)(5^{6n}) = 4(4^{3n}) = 4(64^n) = 4(1)^n = 4$ 
$A_n = 1 + 2 +  4 = 7 = 0$ 
So any of them is divisible by $7$, so it's $7$.
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