Number Theory, Squares in $mathbb{Z}_p ^*,;$ for odd prime $p$

Let $p$ be an odd prime and $\mathrm{gcd}(n, p) = 1.$ Assume that $n = p_1^{a_1} p_2^{a_2} ... p_k ^{a_k}$ is the prime factorization of n. Prove
$$\left(\frac{n}{p}\right) = \left(\frac{p_1}{p}\right)^{i_1} \left(\frac{p_2}{p}\right)^{i_2} ... \left(\frac{p_k}{p}\right)^{i_k},$$ where $i_j = 1$ if $a_j$ is odd, and $i_j = 0$ if $a_j$ is even.