<s>[INST] check if large number $(9^{81}+6)$ is divisible by $11$ I would like to know if there is a mathematical way to check whether number $9^{81}+6$ is divisible by $11$, without actually calculating the whole number.
 [/INST] $$9^1 \equiv -2 \bmod{11}$$ $$9^2 \equiv 4 \bmod{11}$$ $$9^6 \equiv 64 \equiv -2 \bmod{11}$$ $$9^{30} \equiv -32 \equiv 1 \bmod{11}$$ $$9^{81} \equiv 9^{30} 9^{30} 9^{6} 9^{6} 9^{6} 9^2 9^1 \equiv (1)(1)(-2)(-2)(-2)(4)(-2) \equiv 64 \equiv 9 \bmod{11}$$ $$9^{81} + 6 \equiv 15 \equiv 4 \bmod{11}.$$
EDIT: I accidentally wrote $-32 \equiv -1 \bmod{11}$ instead of $-32 \equiv 1 \bmod{11}$. The minus signs fortuitously cancelled!
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