Every positive integer is either nice or naughty, and the Oracle of Numbers knows which are which. However, the Oracle will not directly tell you whether a number is nice or naughty. The only questions the Oracle will answer are questions of the form “What is the sum of all nice divisors of n ?,” where n is a number of the questioner’s choice. For instance, suppose (just for this example) that 2 and 3 are nice, while 1 and 6 are naughty. In that case, if you asked the Oracle, “What is the sum of all nice divisors of 6?,” the Oracle’s answer would be 5.
Show that for any given positive integer n less than 1 million, you can determine whether n is nice or naughty by asking the Oracle at most four questions.