Hello,

I propose an improvement of the answer proposedby Kevin Buzzard as there is no need to factorize.

```
def min_modulus(residues, n):
# clean the input
n = ZZ(n)
residues = sorted(set(ZZ(r)%n for r in residues))
# find the list of number s so that:
# residues + s = residues modulo n
r0 = residues[0]
S = []
for r in residues[1:]:
if all((rr+r-r0)%n in residues for rr in residues):
S.append(r-r0)
# return the answer
if not S:
return residues, n
else:
m = gcd(S)
return sorted(set(r%m for r in residues)), m
```

Then you got

```
sage: min_modulus([0,1,2,3,4,5,6,7], 8)
([0], 1)
sage: min_modulus([0,2,4,6,8], 8)
([0], 2)
sage: min_modulus([1,3,5,7], 8)
([1], 2)
sage: min_modulus([0,4], 8)
([0], 4)
sage: min_modulus([1,5], 8)
([1], 4)
```