MathDB

1

Part of 2022 Switzerland Team Selection Test

Problems(1)

Bases and divisibility

Source: 2022 Switzerland IMO TST, Problem 1

8/1/2022
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
number theoryDivisibilitynumber basesSwitzerland TST