[carpetpool]

So just looking on the conditions mod 3 we have:

a = 1
b = 2
c = 1

a = 0
b = 0
c = 1

I also thought about another exception: n may also define properties of the cyclotomic field K if and only if each prime power p^k dividing n is either 0 or 1 mod n.

So in this sense, factors such as 2^(2*n), 5^(2*n), 11^(2*n), 17^(2*n), 23^(2*n), 29^(2*n)..., etc. would be allowed, but I don't know weather this would make problem harder.

Depending on (prime) K, we also allow n to divide:

for K = 5, 2^(4*n), 3^(4*n), 7^(4*n), 13^(4*n), 17^(4*n), 19^(2*n), 23^(4*n), 29^(2*n)..., etc.

for K = 7, 2^(3*n), 3^(6*n), 5^(6*n), 11^(3*n), 13^(2*n), 17^(6*n), 19^(6*n), 23^(3*n)..., etc.

for K = 11, 2^(10*n), 3^(5*n), 5^(5*n), 7^(10*n), 13^(10*n), 17^(10*n), 19^(10*n), 29^(10*n)..., etc.

for K = 13, 2^(12*n), 3^(3*n), 5^(4*n), 7^(12*n), 11^(12*n), 17^(6*n), 19^(12*n), 23^(6*n), 29^(4*n)..., etc.

and so on...