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PUZZLET 110

Repeated Digit Sums and 15


Take any integer up to 1,000. Add the sum of the digits of its factors, including 1 and itself, to form a new integer. Repeat the process with the new integer until its value is 15.

What is the integer which takes the greatest number of steps to get to 15, and how many steps does it take?

Note: the factors of 12 are 1, 2, 3, 4, 6, and 12, and these are called the Proper Divisors. The digit sum of these factors is: 1 + 2 + 3 + 4 + 6 + (1 + 2) = 19.


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