Let K(a) denote the Kloosterman sum on F3m. It is easy to see that K(a)=2 (mod 3) for all a in F3m. We completely characterize those a in F3m for which K(a)=1 (mod 2), K(a)=0 (mod 4) and K(a)=2 (mod 4). The simplicity of the characterization allows us to count the number of the a in F3m belonging to each of these three classes. As a byproduct we offer an alternative proof for a new class of quasi-perfect ternary linear codes recently presented by Danev and Dodunekov.
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