If r1 rp1 is a reduced residue system modulo a prime P prov

If r_1, ..., r_p-1 is a reduced residue system modulo a prime P, prove that the product of the r_i\'s is congruent to -1 modulo p.

Solution

r1,..,rp-1 form a reduced residue system modulo p, a prime number

Hence they are equal to

1,...,p-1 modulo prime ,p (not in the same order)

Hence, r1*r2*....r_{p-1}=1*...*(p-1)=(p-1)! modulo p

By Wilson\'s Theorem

(p-1)!=-1 modulo p

Hence proved