Past exam of the short options course of the University of Oxford / 2021 / 1 / d / Solution

Complete the square to obtain the equivalent quadratic congruenceHere $\gcd (1111,8093)=1$: the Euclidean algorithm has successive nonzero remainders $316,163,153,10,3,1$. If this modular congruence had a solution, the Jacobi symbol $(1111/8093)$ would be $1$, since a square coprime to an odd modulus has Legendre symbol $1$ at every prime factor. Calculate instead, using Jacobi reciprocity and reduction of numerators,Therefore there is no solution. This obstruction does not require a separate primality test for $8093$; the Jacobi symbol suffices.