The locally exhaustive testing of multiple output combinational circuits is the test which provides exhaustive test patterns for each set of inputs on which each output depends. First, this paper presents a sufficient condition under which a minimum test set (MLTS) for the locally exhaustive testing has 2w test patterns, where w is the maximum number of inputs on which any output depends. Next, we clarify that any CUT with up to four outputs satisfies the condition, independently of w and n, where n is the number of inputs of the CUT. Finally, we clarify that any CUT with five outputs also satisfies the condition for 1w2 or n2wn.