There was a question in a past exam paper that asked Find criteria $\mod(20)$ for determining the Legendre symbol $(\tfrac{-5}{p})$ where $p\geq 7$.
I am very confused by what this means as I understand it the Legendre symbol is implicitly working in $\mod p$, how can we learn anything about $\mod 20$
From Hagen von Eitzen's Comment I believe the following is the correct method :
We're looking at all $p \geq7,$ in $\mod 20$ so then we have to consider $7,11,13,17,19$
say $p\equiv 7$ then we compute $(\tfrac{-5}{7})$
$$(\tfrac{-5}{7})=(\tfrac{2}{7})=1$$ as $7\equiv-1 \mod8$
Now consider $p\equiv11$
$$(\tfrac{-5}{11})=(\tfrac{6}{11})=(\tfrac{2}{11})(\tfrac{3}{11})$$
Note that $11\equiv3 \mod 8$ so we have
$$-(\tfrac{3}{11})=(\tfrac{11}{3})=(\tfrac{2}{3})$$
Where we took the first step here because by the quadratic law of reciprocity $(\tfrac{3}{11})(\tfrac{11}{3})=(-1)^{5\times1}=-1\Rightarrow (\tfrac{3}{11})=-(\tfrac{11}{3})$
$$(\tfrac{2}{3})=-1$$
as $3\equiv 3\mod 4$.
Now we just follow the same method outlined above for $p=13,17,19$and we're done!