Show that the numbers 13 9 4 1 9 18 21 form a complete resid

Show that the numbers -13, -9, -4, -1, 9, 18, 21 form a complete residue system modulo 7.

Solution

-13 = -13 modulo 7 = 7 -6 = 1

-9 = -9 modulo 7 = 7 - 2 = 5

-4 = -4 modulo 7 = 7 -4 = 3

-1 = -1 modulo 7 = 7-1 = 6

9 = 9 modulo 7 = 2

18 = 18 modulo 7 = 4

21 = 21 modulo 7 = 0

So, the given numbers form a complete residue system modulo 7.