{"id":187759,"date":"2023-05-02T22:44:13","date_gmt":"2023-05-03T01:44:13","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=187759"},"modified":"2023-05-21T00:30:30","modified_gmt":"2023-05-21T03:30:30","slug":"problemao-algarismos-9-s","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problemao-algarismos-9-s\/","title":{"rendered":".Problem\u00e3o: Algarismos 9"},"content":{"rendered":"

✏ Link do problema para dispositivos da Apple<\/em>.<\/span><\/a><\/strong><\/p>\n

\n

Problema
\n (Indicado a partir do 7\u00ba ano do E. F.)<\/span><\/p>\n

\n(XXV OBM<\/strong>, Segunda Fase – Adaptado) <\/span> Quantas vezes aparece o algarismo [tex]9[\/tex] no resultado de [tex]\\,\\boxed{10^{100} \u2212 2020}\\,[\/tex]?\n<\/div>\n

\n

\n

Solu\u00e7\u00e3o<\/p>\n

\nVeja que <\/p>\n

\n

[tex]\\qquad\\begin{align}
\n10^{100}-2020 &= 1\\underbrace{00\\cdots 0}_{100 \\text{ zeros}}-(1+2019)\\\\
\n&=1\\underbrace{00\\cdots 0}_{100 \\text{ zeros}}-1-2019\\\\
\n&=\\underbrace{99\\cdots 9}_{100 \\text{ noves}}-2019\\\\
\n&=\\underbrace{99\\cdots 9}_{96 \\text{ noves}}9999-2019\\,.
\n\\end{align}[\/tex]<\/p>\n

Observe a subtra\u00e7\u00e3o [tex]\\underbrace{99\\cdots 9}_{96 \\text{ noves}}9999-2019[\/tex]:<\/p>\n

[tex]\\qquad \\qquad ~~~\\begin{array}{rr}
\n&\\overbrace{99\\cdots 9}^{96 \\text{ noves}}\\,9\\,9\\,9\\,9\\\\
\n– &2\\,0\\,1\\,9\\\\
\n\\hline
\n&\\textcolor{red}{?????????~~~~~~}
\n\\end{array}[\/tex]<\/p>\n

\n

[tex]\\qquad \\qquad ~~~\\begin{array}{rr}
\n&99\\cdots9\\,9\\,9\\,9\\,9\\\\
\n– &2\\,0\\,1\\,9\\\\
\n\\hline
\n&\\textcolor{red}{9\\,9\\cdots9\\,7\\,9\\,8\\,0}
\n\\end{array}[\/tex]<\/p>\n

\n

[tex]\\qquad \\qquad ~~~\\begin{array}{rr}
\n&\\overbrace{99\\cdots 9}^{96 \\text{ noves}}\\,9\\,9\\,9\\,9\\\\
\n– &2\\,0\\,1\\,9\\\\
\n\\hline
\n&\\textcolor{red}{\\underbrace{99\\cdots 9}_{96 \\text{ noves}}\\,7\\,9\\,8\\,0}
\n\\end{array}[\/tex]<\/p>\n

\n

Assim, segue que:
\n[tex]\\qquad\\begin{align}
\n10^{100}-2020 &= \\underbrace{99\\cdots 9}_{96 \\text{ noves}}9999-2019\\\\
\n&=\\textcolor{red}{\\underbrace{99\\cdots 9}_{96 \\text{ noves}}7980}
\n\\end{align}[\/tex]
\ne, portanto, o algarismo [tex]9[\/tex] aparece [tex]97[\/tex] vezes na diferen\u00e7a [tex]\\,\\boxed{10^{100} \u2212 2020}\\,[\/tex].<\/p>\n

\n

Solu\u00e7\u00e3o elaborada pelos Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n

\n","protected":false},"excerpt":{"rendered":"

✏ Link do problema para dispositivos da Apple. Problema (Indicado a partir do 7\u00ba ano do E. F.) (XXV OBM, Segunda Fase – Adaptado) Quantas vezes aparece o algarismo [tex]9[\/tex] no resultado de [tex]\\,\\boxed{10^{100} \u2212 2020}\\,[\/tex]? Solu\u00e7\u00e3o 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 … <\/p>\n

Continue lendo<\/a><\/p>\n","protected":false},"author":254,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-187759","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/187759","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/users\/254"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=187759"}],"version-history":[{"count":30,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/187759\/revisions"}],"predecessor-version":[{"id":189341,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/187759\/revisions\/189341"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=187759"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}