Geometric Sum Modulo a Prime

We first note that $q + q^{2} + .... + q^{772} = q(1 + q + ... + q^{771}) = q \times \dfrac{q^{772} - 1}{q - 1}$ . Thus we are looking for $1 \lt q \lt 773$ such that $q(q^{772} - 1) \equiv 0 \pmod{773}$ .

But since $773$ is prime, by Fermat's Little Theorem we know that $q^{772} \equiv 1 \pmod{773} \Longrightarrow q^{772} - 1 \equiv 0 \pmod{773} \Longrightarrow q(q^{772} - 1) \equiv 0 \pmod{773}$

for all $q$ from $2$ to $772$ , and so the answer is $772 - 2 + 1 = \boxed{771}$ .