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Problema
(Indicado a partir do 7º ano do E. F.)
(XXV OBM, Segunda Fase – Adaptado) Quantas vezes aparece o algarismo [tex]9[/tex] no resultado de [tex]\,\boxed{10^{100} − 2020}\,[/tex]?
Solução
Veja que
[tex]\qquad\begin{align}
10^{100}-2020 &= 1\underbrace{00\cdots 0}_{100 \text{ zeros}}-(1+2019)\\
&=1\underbrace{00\cdots 0}_{100 \text{ zeros}}-1-2019\\
&=\underbrace{99\cdots 9}_{100 \text{ noves}}-2019\\
&=\underbrace{99\cdots 9}_{96 \text{ noves}}9999-2019\,.
\end{align}[/tex]
Observe a subtração [tex]\underbrace{99\cdots 9}_{96 \text{ noves}}9999-2019[/tex]:
[tex]\qquad \qquad ~~~\begin{array}{rr}
&\overbrace{99\cdots 9}^{96 \text{ noves}}\,9\,9\,9\,9\\
– &2\,0\,1\,9\\
\hline
&\textcolor{red}{?????????~~~~~~}
\end{array}[/tex]
[tex]\qquad \qquad ~~~\begin{array}{rr}
&99\cdots9\,9\,9\,9\,9\\
– &2\,0\,1\,9\\
\hline
&\textcolor{red}{9\,9\cdots9\,7\,9\,8\,0}
\end{array}[/tex]
[tex]\qquad \qquad ~~~\begin{array}{rr}
&\overbrace{99\cdots 9}^{96 \text{ noves}}\,9\,9\,9\,9\\
– &2\,0\,1\,9\\
\hline
&\textcolor{red}{\underbrace{99\cdots 9}_{96 \text{ noves}}\,7\,9\,8\,0}
\end{array}[/tex]
Assim, segue que:
[tex]\qquad\begin{align}
10^{100}-2020 &= \underbrace{99\cdots 9}_{96 \text{ noves}}9999-2019\\
&=\textcolor{red}{\underbrace{99\cdots 9}_{96 \text{ noves}}7980}
\end{align}[/tex]
e, portanto, o algarismo [tex]9[/tex] aparece [tex]97[/tex] vezes na diferença [tex]\,\boxed{10^{100} − 2020}\,[/tex].
Solução elaborada pelos Moderadores do Blog.