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Bases and divisibility

Source: 2022 Switzerland IMO TST, Problem 1

August 1, 2022
number theoryDivisibilitynumber basesSwitzerland TST

Problem Statement

Let nn be a positive integer. Prove that there exists a finite sequence SS consisting of only zeros and ones, satisfying the following property: for any positive integer d≥2d \geq 2, when SS is interpreted in base dd, the resulting number is non-zero and divisible by nn. Remark: The sequence S=sksk−1⋯s1s0S=s_ks_{k-1} \cdots s_1s_0 interpreted in base dd is the number ∑i=0ksidi\sum_{i=0}^{k}s_id^i
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