Question 27
Determine all sequences of positive integers $\{a_n\}_{n=1}^{\infty}$ satisfying the following conditions for any positive integer $k$:
(i) $a_{2^{k+1}} = 2 \cdot a_{2^k}$,
(ii) The set $\{a_1, a_2, \dots, a_k\}$ is a complete set of residue classes modulo $k$.
(i) $a_{2^{k+1}} = 2 \cdot a_{2^k}$,
(ii) The set $\{a_1, a_2, \dots, a_k\}$ is a complete set of residue classes modulo $k$.