MathDB

5

Part of 2021 Mexico National Olympiad

Problems(1)

$n=\overline{a_1a_2\cdots a_{k-1}a_k}$

Source: Mexico National 2021 P5

11/11/2021
If n=a1a2⋯ak−1ak‾n=\overline{a_1a_2\cdots a_{k-1}a_k}, be s(n)s(n) such that . If kk is even, s(n)=a1a2‾+a3a4‾⋯+ak−1ak‾s(n)=\overline{a_1a_2}+\overline{a_3a_4}\cdots+\overline{a_{k-1}a_k} . If kk is odd, s(n)=a1+a2a3‾⋯+ak−1ak‾s(n)=a_1+\overline{a_2a_3}\cdots+\overline{a_{k-1}a_k} For example s(123)=1+23=24s(123)=1+23=24 and s(2021)=20+21=41s(2021)=20+21=41 Be nn is digitaldigital if s(n)s(n) is a divisor of nn. Prove that among any 198 consecutive positive integers, all of them less than 2000021 there is one of them that is digitaldigital.
number theory