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Math Prize 2017 Problem 9

Source:

September 26, 2017
Math Prize for Girls

Problem Statement

Say that a positive integer nn is smooth if 1n\frac{1}{n} has a terminating decimal expansion. (Note that 1 is smooth.) Compute the value of the infinite series ∑n1n3 , \sum_n \frac{1}{n^3} \, , where nn ranges over all smooth positive integers.
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