Question

Show that if p is an odd prime and a and b are integers with $a\equiv b(modp)$, then$\left(\frac{a}{p}\right)=\left(\frac{b}{p}\right)$

Short answer

If p is an odd prime and a and b are integers with $a\equiv b(modp)$, then $\left(\frac{a}{p}\right)=\left(\frac{b}{p}\right)$

Step-by-step solution

Step: 1

The value of $\left(\frac{a}{p}\right)$depends only on whether a is a quadratic residue modulo p. That is, whether $x^2\equiv a(modp)$has a solution. Because this depends only on the equivalence class of a modulo p, it follows that $\left(\frac{a}{p}\right)=\left(\frac{b}{p}\right)$if$a\equiv b(modp)$