Determine the least non-negative residues $x$ in the following:
(a) $x \equiv 1 + 2 + 2^2 + 2^3+2^4 + 2^5 \pmod 7$
(b) $x \equiv 1 + 3 + 3^2 + 3^3+3^4 + 3^5 \pmod 7$
(c) $x \equiv 1 + 3 + 3^2 + 3^3 + 3^4 + 3^5+3^6 + 3^7 + 3^8 + 3^9 \pmod {11}$
(d) $x \equiv 1 + 2 + 2^2 + 2^3 + 2^4 + 2^5+2^6 + 2^7 + 2^8 + 2^9 + 2^{10} + 2^{11} \pmod {13}$
What do you notice about your results?
(a) We have
$$ \begin{align} x & \equiv 1 + 2 + 2^2 + 2^3+2^4 + 2^5 \pmod 7 \\ \\ & \equiv 1 + 2 + 4 + 1 + 2 + 4 \pmod 7 \\ \\ & \equiv 7 + 7 \pmod 7 \\ \\ & \equiv 0 + 0 \pmod 7 \\ \\ & \equiv 0 \pmod 7 \end{align} $$
(b) We have
$$ \begin{align} x & \equiv 1 + 3 + 3^2 + 3^3+3^4 + 3^5 \pmod 7 \\ \\ & \equiv 1 + 3 + 2 + 6 + 4 + 5 \pmod 7 \\ \\ & \equiv (6+1) + (5+2) + (4+3) \pmod 7 \\ \\ & \equiv 0 + 0 +0 \pmod 7 \\ \\ & \equiv 0 \pmod 7 \end{align} $$
(c) We have
$$ \begin{align} x & \equiv 1 + 3 + 3^2 + 3^3 + 3^4 + 3^5+3^6 + 3^7 + 3^8 + 3^9 \pmod {11} \\ \\ & \equiv 1 + 3 + 9 + 5 + 4 + 1 + 3 + 9 + 5 + 4 \pmod {11} \\ \\ & \equiv (9 +1 + 1) + (3+3+5) + (4+9 + 5+4) \pmod {11} \\ \\ & \equiv 11 + 11 + 22 \pmod {11} \\ \\ & \equiv 0 \pmod {11} \end{align} $$
(d) We have
$$ \begin{align} x & \equiv 1 + 2 + 2^2 + 2^3 + 2^4 + 2^5+2^6 + 2^7 + 2^8 + 2^9 + 2^{10} + 2^{11} \pmod {13} \\ \\ & \equiv 1 + 2 + 4 + 8 + 3 + 6 + 12 + 11 + 9 + 5 + 10 + 7 \pmod {13} \\ \\ & \equiv (12 + 1) + (11 + 2) + (8+5) + (9+4) + (10+3) + (6+7) \pmod {13} \\ \\ & \equiv 0 + 0 + 0 + 0 + 0 +0 \pmod {13} \\ \\ & \equiv 0 \pmod {13} \end{align} $$
All the results are 0 modulo the given prime.