What proportion of all the positive integers are a multiple of their first digit?

(*Credits*: I learnt of this problem from James Tanton’s Twitter feed.)

Hint 1: Break up the positive integers into groups based on what their first digit is. For each of these groups, can you find the proportion which are a multiple of their first digit?

Scroll down for Hint 2…

Hint 2: For each group, break it up further based on the number of digits. For example, for all positive integers starting with the number ‘2’, break it up as follows: {2}, {20, 21, …, 29}, {200, 201, …, 299}, … For each group, what proportion are a multiple of the number 2?

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