Determine the set of primes for which 11 is a quadratic residue.
First, note that 11 is a quadratic residue mod 2 (e.g. $\displaystyle 1^2\equiv11\,(\bmod\,2)).$
Let $\displaystyle p$ and $\displaystyle q$ be odd primes. There is a result (equivalent to the law of quadratic reciprocity) that says that $\displaystyle (11\,|\,p)=(11\,|\,q)$ if and only if $\displaystyle p\equiv\pm q\,(\bmod\,44).$ So you only have to determine all odd primes $\displaystyle p < 44$ for which $\displaystyle (11\,|\,p)=1$ All other odd primes for which 11 is a quadratic residue will congruent to plus or miinus one of these.
I came across it in H.E. Rose, A Course in Number Theory, p.67:
Theorem 2.2 Let $\displaystyle p$ and $\displaystyle q$ be distinct odd primes and $\displaystyle a\ge1.$ The quadratic reciprocity law is equivalent to
if $\displaystyle p\equiv\pm q\,(\bmod\,4a)$ then $\displaystyle (a/p)=(a/q).$