Evaluate the Legendre Symbols (503/773)

Question

Evaluate the Legendre Symbols (503/773)

Solution: (503/773) = (270/503) = (2/503)(3^3/503)(5/503) = 1*(5/503)(3/503) = (503/5)(-1)(503/3) = -(3/5)(2/3) = -1

I don't understand how they obtain (-(3/5)(2/3)) from (503/5)(-1)(503/3).

Answer 1

Note that $503\equiv 3$ (mod $5$).

Obviously, both $\displaystyle \left(\frac{503}{5}\right)$ and $\displaystyle \left(\frac{3}{5}\right)$ are not zero.

If $\displaystyle \left(\frac{503}{5}\right)=1$, then $\exists x\in\mathbb{Z}$ such that $x^2\equiv503$ (mod $5$). This implies that $x^2\equiv3$ (mod $5$).

Conversely, if $\displaystyle \left(\frac{3}{5}\right)=1$, then $\exists y\in\mathbb{Z}$ such that $y^2\equiv503$ (mod $5$). This implies that $y^2\equiv503$ (mod $5$).

So $\displaystyle \left(\frac{503}{5}\right)=\left(\frac{3}{5}\right)$.

As $503\equiv 2$ (mod $3$), $\displaystyle \left(\frac{503}{3}\right)=\left(\frac{2}{3}\right)$ by similar argument.