<s>[INST] Which of the following sets is a complete set of representatives modulo 7? Which of the following sets is a complete set of representatives modulo 7?
1) (1, 8, 27, 64, 125, 216, 343)
1 mod 7 = 1
8 mod 7 = 1
27 mod 7 = 6
64 mod 7 = 1
125 mod 7 = 6
216 mod = 6
343 mod = 0
2) (1, -3, 9, -27, 81, -243, 0)
1 mod 7 = 1
-3 mod 7 = 4
9 mod 7 = 2
-27 mod 7 = 1
81 mod 7 = 4
-243 mod 7 = 2
0 mod 7 = 0
3) (0, 1, -2, 4, -8, 16, -32)
0 mod 7 = 0
1 mod 7 = 1
-2 mod 7 = 5
4 mod 7 = 4
-8 mod 7 = 6
16 mod 7 = 2
-32 mod 7 = 3
i think its only 1. but wanted to make sure.
 [/INST] Hint: you want a complete set whose members are equivalent, modulo 7, to one and only one element of the set: $\{0, 1, 2, 3, 4, 5, 6\}$, not necessarily in that order. Each and every one of these numbers must be represented by one and only one of the numbers in such a set. That is, you want representatives of the equivalence classes, modulo 7 which will consist of seven distinct integers, each one of which can be expressed in one and only one of the following  forms $$(7k_1, 7k_2 + 1, 7k_3 +2, 7k_4 + 3, 7k_5 + 4, 7k_6 +5, 7k_7 + 6), \quad k_i \in \mathbb Z)$$
$(1)$ is not such a set.
Added
Note that the elements of $(3)$ represent each equivalence class, modulo 7, and hence forms a complete set of representatives: $$ 0\equiv 0 \pmod 7,\quad  \;1 \equiv 1 \pmod 7, \quad -2\equiv 5\pmod 7,\quad \\ 4\equiv 4 \pmod 7,\quad  -8 \equiv 6 \pmod 7,\quad 16\equiv 2 \pmod 7,\quad -32 \equiv 3 \pmod 7$$
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