The set A is low for Martin-Lof random if each random set is already random relative to A. A is K-trivial if the prefix complexity K of each initial segment of A is minimal, namely K(n)+O(1). We show that these classes coincide. This implies answers to questions of Ambos-Spies and Kucera [2], showing that each low for Martin-Lof random set is # 2 . Our class induces a natural intermediate # 3 ideal in the r.e. Turing degrees (which generates the whole class under downward closure). Answering