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Problema Adaptado: UFRGS<\/strong> – 2015.<\/p>\n<\/div>\n \n Solu\u00e7\u00e3o<\/p>\n Solu\u00e7\u00e3o elaborada pelos Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n \n","protected":false},"excerpt":{"rendered":" Problema (Indicado a partir do 8\u00ba ano do E. F.) Encontre o algarismo das unidades do n\u00famero [tex]7^{77}-3^{33}[\/tex]. Adaptado: UFRGS – 2015. Solu\u00e7\u00e3o Observe que [tex]\\qquad \\qquad \\begin{equation} \\begin{split} 3^1 &= \\textcolor{blue}{3}\\\\ 3^2 &= \\textcolor{red}{9}\\\\ 3^3 &= 2\\textcolor{orange}{7}\\\\ 3^4 &= 8\\textcolor{green}{1}\\\\ 3^5 &= 24\\textcolor{blue}{3}\\\\ 3^6 &= 72\\textcolor{red}{9}\\\\ \\vdots \\end{split} \\end{equation}\\\\ \\,[\/tex] e perceba que as … <\/p>\n
\n (Indicado a partir do 8\u00ba ano do E. F.)<\/span><\/p>\n
\nEncontre o algarismo das unidades do n\u00famero [tex]7^{77}-3^{33}[\/tex].<\/p>\n
\nObserve que
\n[tex]\\qquad \\qquad
\n\\begin{equation}
\n\\begin{split}
\n3^1 &= \\textcolor{blue}{3}\\\\
\n3^2 &= \\textcolor{red}{9}\\\\
\n3^3 &= 2\\textcolor{orange}{7}\\\\
\n3^4 &= 8\\textcolor{green}{1}\\\\
\n3^5 &= 24\\textcolor{blue}{3}\\\\
\n3^6 &= 72\\textcolor{red}{9}\\\\
\n\\vdots
\n\\end{split}
\n\\end{equation}\\\\
\n\\,[\/tex]
\ne perceba que as pot\u00eancias cujos expoentes s\u00e3o m\u00faltiplos de [tex]4[\/tex] possuem algarismo das unidades igual a [tex]1[\/tex]. Como [tex]32 = 4\\times 8[\/tex], ent\u00e3o o algarismo das unidades de [tex]3^{32}[\/tex] \u00e9 [tex]1[\/tex]. Portanto, [tex]3^{33}[\/tex] tem o \u00faltimo algarismo igual a [tex]3[\/tex], j\u00e1 que [tex]3^{33}=3^{32}\\times 3\\,.[\/tex]
\nAgora, veja que
\n[tex]\\qquad \\qquad
\n\\begin{equation}
\n\\begin{split}
\n7^1 &= \\textcolor{blue}{7}\\\\
\n7^2 &= 4\\textcolor{red}{9}\\\\
\n7^3 &= 34\\textcolor{orange}{3}\\\\
\n7^4 &= 2\\;40\\textcolor{green}{1}\\\\
\n7^5 &= 16\\;80\\textcolor{blue}{7}\\\\
\n7^6 &= 117\\;64\\textcolor{red}{9}\\\\
\n\\vdots
\n\\end{split}
\n\\end{equation}\\\\
\n\\,[\/tex]
\nDe maneira an\u00e1loga, podemos ver que as pot\u00eancias cujos expoentes s\u00e3o m\u00faltiplos de [tex]4[\/tex] possuem algarismo das unidades igual a [tex]1[\/tex]. Como [tex]76 = 4\\times 19[\/tex], ent\u00e3o o algarismo das unidades de [tex]7^{76}[\/tex] \u00e9 [tex]1[\/tex] e portanto, [tex]7^{77}[\/tex] tem o \u00faltimo algarismo igual a [tex]7[\/tex], pois [tex]7^{77}=7^{76} \\times 7\\,.[\/tex]
\nLogo, o algarismo das unidades de [tex]7^{77}-3^{33}[\/tex] \u00e9 [tex]\\,\\fcolorbox{black}{#eee0e5}{$7-3 = 4$}\\,.[\/tex]<\/p>\n
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