Prove that $r_1s_1, r_2s_2, ..., r_{p-1}s_{p-1}$ is not a reduced residue system modulo p.

Question

If $r_1, ... r_{p-1}$ and $s_1, ... s_{p-1}$ are two reduced residue systems modulo an odd prime p, prove that $r_1s_1, r_2s_2, ..., r_{p-1}s_{p-1}$ is not a reduced residue system modulo p.

don't know where to start, any help appreciated!

Answer 1

We can permute the $r$-values so that the $r$-system is exactly $1,2,3,\dots,(p-1)$. Using a theorem of Wilson, the product of the elements in the given systems are modulo $p$:

$$ (-1)=(p-1)!=\prod_k r_k=\prod_k s_k\mod p\ , $$ but $$ \prod_k (r_ks_k)=\left(\prod_k r_k\right)\left(\prod_k s_k\right)=(-1)(-1)=1\mod p\ , $$ so $(r_ks_k)_k$ is not a reduced system modulo p.