For $n=pq$, where p,q are distinct odd primes show that there exist $\alpha, \beta \in Z_n^*$ such that $\alpha, \beta \notin (Z_n^*)^2$ and $\alpha \cdot \beta \notin (Z_n^*)^2$.
Here if we try to use Chinese Remainder theorem results then let :
$$\alpha \equiv \alpha_p\mod p\,\,,\qquad\alpha \equiv \alpha_q\mod q$$
$$\beta \equiv \beta_p\mod p\,\,,\qquad\beta \equiv \beta_q\mod q$$
Since $\alpha \notin (Z_n^*)^2$, what can we say about $\alpha_p$ and $\alpha_q$? Do $\alpha_p \notin (Z_p^*)^2$ and $\alpha_q \notin (Z_q^*)^2$ also hold? Is there another approach to solve this problem?
PS: Couldn't insert "p,q are distinct primes" in title due length restriction.