A natural integer N can be written as x mod p, for instance:
N... 1 2 3 4 5 6 7 8 9 10 11 12
p=2 1 2 1 2 1 2 1 2 1 2 1 2
p=3 1 2 3 1 2 3 1 2 3 1 2 3
p=5 1 2 3 4 5 1 2 3 4 5 1 2
p=7 1 2 3 4 5 6 7 1 2 3 4 5
Etc.
The puzzle is quite difficult to state in this format but I'll have a go:
You...