ISBNs...

Level pending

An ISBN ( International Stanadard Book Number) is a ten digit code which uniquely identifies a book. The first nine digits represent the Group, Publisher and Title of the book and the last digit is used to check whether ISBN is correct or not.

Each of the first nine digits of the code can take a value between \(0\) and \(9\) (inclusive). Sometimes it is necessary to make the last digit equal to ten; this is done by writing the last digit of the code as \(X\). To verify an ISBN, calculate \(10\) times the first digit, plus \(9\) times the second digit, plus \(8\) times the third and so on until we add \(1\) time the last digit. If the final number leaves no remainder when divided by \(11\), the code is a valid ISBN.

For example:

\(02011003311 = 10*0 + 9*2 + 8*0 + 7*1 + 6*1 + 5*0 + 4*3 + 3*3 + 2*1 + 1*1 = 55\) Since \(55\) leaves no remainder when divisible by \(11\), hence it is a valid ISBN.

The problem: Let \(\xi\) be the prime factorization of the number of valid ISBN codes possible. Find the sum of powers of the prime factors in \(\xi\).

Details and assumptions

If the number of codes would have been \(50\), then \(\xi\) would have been \(2 * 5^2\) and hence the answer would have been \(2+1=3\). However, \(3\) is not the answer. :(

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