Greetings! The problem: if r1, r2, r3, ..., rp and r'1, r'2, r'3, ... , r'p are any two complete residue systems modulo a prime p > 2, prove that the set r1 * r'1, r2 * r'2, r3 * r'3, ... , rp * r'p cannot be a complete residue system modulo p.
Greetings! The problem: if r1, r2, r3, ..., rp and r'1, r'2, r'3, ... , r'p are any two complete residue systems modulo a prime p > 2, prove that the set r1 * r'1, r2 * r'2, r3 * r'3, ... , rp * r'p cannot be a complete residue system modulo p.
Oh sorry I was thinking reduced system. I hadn't had my coffee yet . But the heart of the solution is there.
One of the 's is (say ) and similarily for one of the 's (say ). So if we want to avoid having two of the 's congruent to 0 - because then we obviously couldn't get a complete system - then and must be paired together. The remaining products will have to form a reduced system.
The argument I gave explains why that's impossible.
Saying that is a complete system of residues is just saying that , where denotes the image of modulo .
Thus for some permutation of we will have :
...
and taking the product of these congruences you get the answer to your question.
The product of the elements of any reduced system of residues mod p is = -1 (mod p). It doesn't matter whether you reduce mod p before or after taking the product.