as for the property about 0 and 1 mod n only as factors we have restrictions for example:

- (3y+1)(3x+1) = 9xy+3y+3x+1
- (3y+1)(3x) = 9xy+3x
- (3y)(3x) = 9xy

all look fine until you realize that x or y in each case has some restriction in number 3 for example, neither x nor y can be of any other remainders except 0 and 1 mod 3 ( mod n respectively). in the second one x has to be of remainder 0 or 1 mod 3 ( mod n respectively), and in the first two cases y can't be odd ( in the first case x can also not be odd).