Additive bijection between module sets.

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This is already been solved, but lacks of what is inteded by additive bijection. Could someone tell me what is the meaning of the map?

(like how would I read that, in the sense of how is the connection between the remainders and to the module of 6) (Also, why was $3^{n}$ used?)

The simpler the explanation the better. Thank you in advance.