Determine the probability that for a positive base ten integer X drawn at random between 1 and 100 inclusively, the number 10X is expressible as the product of precisely two positive integers, neither of which contains the digit zero.
Before I read Charlie's comment, I came up with the same result of probability = 0.1 or 10% using Windows Calculator in scientific mode. I let set A = {n ‾ 100 | 2明 contains no zeros} and set B = {n ‾ 100 | 5明 contains no zeros}, with the goal of finding the size of the set A目B (A intersected with B) and dividing by 100. I found A = {1-9, 13-16, 18, 19, 24, 25, 27, 28, 31-37, 39, 49, 51, 67, 72, 76, 81, 86}. But due to size limitations was only able to determine the smaller elements of B with absolute certainty, which included 1-7, 9-11, 17, 18, 30, and 33. For n > 45, the result for 5明 had to be displayed in scientific notation. However the subset A' = {45 < n ‾ 100} = {49, 51, 67, 72, 76, 81, 86}, so it only remained to check these seven numbers to NOT be elements of B, each of which luckily showed zeros early enough in the scientific notation. Therefore A目B = {1-7, 9, 18, 33} with a size of 10, giving the solution above.
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