Question

Determinati restul impartirii numarului n=201999...9 (de 2020 de ori 9) la 101

Answer 1

$\displaystyle\it\\observam~ca~2019=\mathcal{M}_{101}+100.\\20199=\mathcal{M}_{101}+100.\\201999=\mathcal{M}_{101}+100.\\....\\ne~propunem~deci~sa~aratam~ca~numarul~n~este~\mathcal{M}_{101}+100.\\201\underbrace{\it 9999999...9}_{de~2020~ori}=201\underbrace{\it 999999...9}_{de~2017~ori}899+100.\\dar,~201\underbrace{\it9999...9}_{de~n~ori}899=1\underbrace{999999...9}_{de~n+3~ori}\cdot101 = \mathcal{M}_{101}.$