Solvability of congruences in number theory.

Question

Prove the following lemma.

Let $p_1,q_1,....,p_r,q_r,m$ be numbers with $gcd(p_1,....,p_r,m)=1$. Then the congruences $$p_i B \equiv q_i\ (\textrm{mod}\ m)\,\,\, ,\,\, i=1,...,r$$ have a unique solution modulo $m$ if and only if for all $i,j=1,......,r$ we have $$p_i q_j \equiv p_j q_i (\textrm{mod}\ m)$$.

I have tried using the congruences formula from the elementary number theory but didn't succeed. I would really appreciate if someone can give me an explicit proof.

Answer 1

$p_iB\equiv q_i\bmod m$ all $i$ implies $B\equiv p_i^{-1}q_i\bmod m$ all $i$, so $p_i^{-1}q_i\equiv p_j^{-1}q_j\bmod m$ all $i$, $j$, and we're done.