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Arithmetic puzzles are Mad Libs for math: fill in the blanks with numbers or operations to make the equation true.

Given a positive integer \(n\), let \(p(n)\) be the product of the non-zero digits of \(n\). (If \(n\) has one digit, then \(p(n)\) is equal to that digit.) Let

\[S = p(1) + p(2) + \cdots + p(999). \]

What is the largest prime factor of \(S\)?

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\[ 12345678910111213141516 \ldots \]

The number above shows a concatenating of the natural numbers in ascending order. What is the \(28383^\text{rd} \) digit from the left (1 being the first) of the number above?

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Krishna always reads some (at least 2) pages of "Harry Potter" before going to school. One good day, Agnishom asked him - "Krishna, what is the sum of all the page numbers you read today?"

Krishna replied "It is either 512 or 412."

What is it?

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For any positive integer \(k\) let \(f_1(k)\) denote the sum of the squares of the digits of \(k\) (when written in decimal), and for \(n \ge 2\) define \(f_n(k)\) iteratively by \(f_n(k)=f_1(f_{nā1}(k))\).

Find \(f_{2017}(2016)\).

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