# 2021 April MIMC 10 Problems/Problem 19

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can be expressed as in base which is a positive integer. Find the sum of the digits of .

## Solution

We realize that when a decimal is expressed in base , the decimal would equal to the expression . Use this idea, .

This sum is basically the sum of two infinite geometric series. The first one has first term of and a common ratio of . The second one has first term and a common ratio of . The total sum is . This would result in . Turn this into a quadratic by cross-multiplication, we would get . HOWEVER, all numbers in base can only have as its digits. Therefore, the answer will be .