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jmc

number theory senior

Problem

If , where represents a base-8 digit (and represents a four-digit number whose second digit is ), then what is the sum of all possible values of in base 10?
Solution
Convert to base 10 to get . Since the possible values of are 0, 1, 2,..., 7, the possible values of form an arithmetic series with first term 545 and last term . The sum of an arithmetic series is , so the sum of the possible values of is .
Final answer
6152