Complete residue system Question

[sty2004]

Hi i am doing self-study of number theory as it looks interesting and enlightening.
Can someone help because I encounter a problem here..
Suppose A = {a₁,a₁,,,,,,,a_k} is a complete residue system modulo k. Prove that for each integer n and each nonnegative integer s there exists a congruence of the form
n ≡ (sum j=0 to s)b_j k^j ( mod k^s+1 )
where b_j\in A for each j .

 

[CRGreathouse]

a_1 and a_k are relatively prime, so you can use the extended Euclidean algorithm to find constants A and B with Aa_1 + Ba_k = 1. Then take (An)a_1 + (Bn)a_k.