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
.