{"id":58443,"date":"2016-03-30T19:41:13","date_gmt":"2016-03-30T22:41:13","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=58443"},"modified":"2020-11-04T01:54:27","modified_gmt":"2020-11-04T03:54:27","slug":"numeros-especiais-numeros-perfeitos-problemas","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/numeros-especiais-numeros-perfeitos\/numeros-especiais-numeros-perfeitos-problemas\/","title":{"rendered":"N\u00fameros especiais \u2013 n\u00fameros perfeitos: Problemas"},"content":{"rendered":"<p style=\"margin:30px;\">\n<div class=\"grie cb92\">\n<div class=\"ti1 c9\" style=\"line-height:20px;margin:10px\"><em> N\u00fameros perfeitos<\/em><\/div>\n<div class=\"ti2 c9\" style=\"line-height:20px;margin:10px\"><em> Problemas<\/em><\/div>\n<hr style=\"height:3px; width:100%;background-color:#6fa34c; margin-top: 10px; margin-bottom: 40px;border: 0 none\"\/>\n<div class=\"gree cb94\" style=\"width:80%;\">\n<strong>Problema 1:<\/strong><br \/>\n<strong>(i)<\/strong> Mostrar que <span class=\"lat\">672<\/span>  \u00e9 um n\u00famero multiperfeito. Qual a sua ordem?<br \/>\n<strong>(ii)<\/strong> Mostrar que <span class=\"lat\">30240<\/span>  \u00e9 um n\u00famero multiperfeito de ordem <span class=\"lat\">4<\/span>.<br \/>\n<strong>(iii)<\/strong> Mostrar que <span class=\"lat\">523776<\/span> \u00e9 um n\u00famero multiperfeito de ordem <span class=\"lat\">3<\/span>.<br \/>\n<strong>(iv)<\/strong> Mostrar que <span class=\"lat\">2178540<\/span> \u00e9 um n\u00famero multiperfeito de ordem <span class=\"lat\">4<\/span>.\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb94\" style=\"width:80%;\">\n<strong>Problema 2:<\/strong> Mostre que um n\u00famero primo n\u00e3o pode ser perfeito.<\/p>\n<div style=\"text-align: center;\">\n<input  class=\"bo3 c9 \" style=\" margin-left: auto; margin-right: auto;\" value=\"Solu\u00e7\u00e3o\" title=\"Clique aqui.\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Solu\u00e7\u00e3o';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<div class=\"gr1 cb93\" style=\"width:90%;\">\n<strong>Solu\u00e7\u00e3o:<\/strong> Seja [tex]p[\/tex] um n\u00famero natural primo qualquer. Como os \u00fanicos divisores naturais de um n\u00famero primo s\u00e3o <span class=\"lat\">1<\/span> e o pr\u00f3prio n\u00famero, ent\u00e3o [tex]\\sigma(p)=p+1[\/tex].<br \/>\nSe [tex]p[\/tex] fosse perfeito, [tex]\\sigma(p)=2p[\/tex] e, ent\u00e3o, [tex]p+1=\\sigma(p)=2p[\/tex]. Dessa forma ter\u00edamos [tex]p+1=2p[\/tex] e [tex]p=1[\/tex], o que n\u00e3o \u00e9 poss\u00edvel, visto que [tex]p[\/tex] \u00e9 primo.<br \/>\nPortanto um n\u00famero primo n\u00e3o pode ser perfeito.\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb94\"  style=\"width:80%;\">\n<strong>Problema 3:<\/strong> <\/p>\n<p class=\"gr3 bg92\" style=\"width:60%;\">\nUm n\u00famero natural n\u00e3o nulo [tex]n[\/tex] \u00e9 dito <strong><em>perfeito multiplicativo<\/em><\/strong> se o produto dos seus divisores for igual a [tex]n^2[\/tex].<\/p>\n<p>Determinar os dez primeiros n\u00fameros naturais perfeitos multiplicativos.\n<\/p><\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb94\"  style=\"width:80%;\">\n<strong>Problema 4:<\/strong> <\/p>\n<p><strong>(i)<\/strong> Seja  [tex]n[\/tex] um n\u00famero multiperfeito de ordem <span class=\"lat\">3<\/span>.<br \/>\nMostre que se [tex]n[\/tex] n\u00e3o for m\u00faltiplo de 3, ent\u00e3o [tex]3n[\/tex] \u00e9  um n\u00famero multiperfeito de ordem <span class=\"lat\">4<\/span>.<\/p>\n<p style=\"margin:20px;\">\n<p><strong>(ii)<\/strong> Seja  [tex]n[\/tex] um n\u00famero multiperfeito de ordem <span class=\"lat\">5<\/span>.<br \/>\nMostre que se [tex]n[\/tex] n\u00e3o for m\u00faltiplo de 5, ent\u00e3o [tex]5n[\/tex] \u00e9  um n\u00famero multiperfeito de ordem <span class=\"lat\">6<\/span>.<\/p>\n<p style=\"margin:20px;\">\n<div style=\"text-align: center;\">\n<input  class=\"bo3 c9 \" style=\" margin-left: auto; margin-right: auto;\" value=\"Solu\u00e7\u00e3o\" title=\"Clique aqui.\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Solu\u00e7\u00e3o';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<div class=\"gr1 cb93\"  style=\"width:90%;\">\n<strong>Solu\u00e7\u00e3o de (i):<\/strong> Seja [tex]n[\/tex] um n\u00famero multiperfeito de ordem <span class=\"lat\">3<\/span>. Assim [tex]n[\/tex] \u00e9 um n\u00famero natural maior do que <span class=\"lat\">1<\/span> e [tex]\\sigma(n)=3n[\/tex]. Agora, observe que:<\/p>\n<ul>\n<li>todo divisor de [tex]n[\/tex] \u00e9 divisor de [tex]3n[\/tex];<\/li>\n<li><span class=\"lat\">3<\/span> \u00e9 primo e n\u00e3o \u00e9 divisor de [tex]n[\/tex];<\/li>\n<\/ul>\n<p>portanto, <\/p>\n<li>se [tex]d_1, \\, d_2, \\, \\cdots, \\, d_r[\/tex] s\u00e3o os divisores de [tex]n[\/tex], ent\u00e3o [tex] d_1, \\, d_2, \\, \\cdots, d_r, \\,  \\, 3\\cdot d_1, \\,  \\, 3\\cdot d_2, \\, \\cdots, \\, 3\\cdot d_r[\/tex] s\u00e3o os divisores de [tex]3n[\/tex].<\/li>\n<p>Assim:<br \/>\n[tex]\\begin{align*} \\quad\\sigma(3n) &#038;=  d_1+d_2+\\cdots +d_r+3\\cdot d_1+3 \\cdot d_2+\\cdots+3\\cdot d_r \\\\ &#038; =\\left( d_1+d_2+\\cdots +d_r \\right)+3\\cdot \\left(d_1+ d_2+\\cdots+ d_r\\right) \\\\ &#038; =\\sigma(n)+3\\cdot \\sigma(n) \\\\&#038; =4\\cdot \\sigma(n).<br \/>\n\\end{align*}[\/tex]<\/p>\n<p style=\"margin:10px;\">\n<p>Mas [tex]\\sigma(n)=3n[\/tex], logo [tex]\\sigma(3n) =12n [\/tex].<br \/>\nDessa forma  [tex]\\sigma([3n]) =4[3n] [\/tex] e, portanto, [tex]3n[\/tex] \u00e9  um n\u00famero multiperfeito de ordem <span class=\"lat\">4<\/span>.\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:20px\">\n<div class=\"gree cb94\"  style=\"width:80%;\">\n<strong>Problema 5:<\/strong> <\/p>\n<p>Prove  que se [tex]n[\/tex] \u00e9 uma pot\u00eancia de um n\u00famero primo, ent\u00e3o [tex]n[\/tex] n\u00e3o \u00e9 um n\u00famero perfeito.<\/p>\n<div style=\"text-align: center;\">\n<input  class=\"bo3 c9 \" style=\" margin-left: auto; margin-right: auto;\" value=\"Dica\" title=\"Clique aqui.\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Dica';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<div class=\"gr1 cb93\" style=\"width:90%;\">\n<ul>\n<li>Verifique que se [tex]n=p^r[\/tex] com [tex]p[\/tex] e [tex]r[\/tex] n\u00fameros naturais n\u00e3o nulos, sendo [tex]p[\/tex] primo, ent\u00e3o os divisores de [tex]n[\/tex] s\u00e3o [tex]1, \\, p, \\, \\cdots, \\, p^r[\/tex].<\/li>\n<li>Efetue a soma  [tex]1+p+p^2+\\cdots+p^r[\/tex].<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:10px;\">\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb94\"  style=\"width:80%;\">\n<strong>Problema 6:<\/strong><br \/>\nProve  que um quadrado perfeito n\u00e3o \u00e9 um n\u00famero perfeito.<\/p>\n<p style=\"margin:10px;\">\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb94\"  style=\"width:80%;\">\n<strong>Problema 7:<\/strong><br \/>\nProve que o produto de dois primos \u00edmpares nunca \u00e9 um n\u00famero perfeito.<\/p>\n<p style=\"margin:20px;\">\n<div style=\"text-align: center;\">\n<input  class=\"bo3 c9 \" style=\" margin-left: auto; margin-right: auto;\" value=\"Sugest\u00e3o\" title=\"Clique aqui.\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Sugest\u00e3o';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<div class=\"gr1 cb93\" style=\"width:90%;\">\n<strong>Solu\u00e7\u00e3o:<\/strong> Mostre, inicialmente, que se [tex]a[\/tex] e [tex]b[\/tex] s\u00e3o n\u00fameros maiores do que <span class=\"lat\">2<\/span> ent\u00e3o [tex]ab\\gt a+b+1[\/tex].<br \/>\nEm seguida, verifique que se [tex]p[\/tex] e [tex]q[\/tex] s\u00e3o primos naturais \u00edmpares e distintos, ent\u00e3o os divisores do n\u00famero [tex]n=pq[\/tex] s\u00e3o [tex]1, \\, p, \\, q, \\, pq[\/tex].<br \/>\nFinalmente, utilize a desigualdade  [tex]pq\\gt p+q+1[\/tex] para garantir que n\u00e3o \u00e9 poss\u00edvel a igualdade [tex]\\sigma(n)=2n[\/tex] para  [tex]n=pq[\/tex], com [tex]p[\/tex] e [tex]q[\/tex] primos naturais \u00edmpares e distintos. (O caso em que [tex]p=q[\/tex] recai no problema anterior.)\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb94\"  style=\"width:80%;\">\n<strong>Problema 8:<\/strong> <\/p>\n<p>Sejam [tex]d_1, \\, d_2, \\, \\cdots \\, , \\, d_r[\/tex] os divisores positivos de um n\u00famero natural [tex]n[\/tex], [tex]n\\gt 1[\/tex].<br \/>\nProve que se [tex]n[\/tex] \u00e9 um n\u00famero perfeito, ent\u00e3o [tex] \\, \\dfrac{1}{d_1}+\\dfrac{1}{d_2}+\\cdots+\\dfrac{1}{d_r}=2[\/tex].<\/p>\n<p style=\"margin:20px;\">\n<div style=\"text-align: center;\">\n<input  class=\"bo3 c9 \" style=\" margin-left: auto; margin-right: auto;\" value=\"Solu\u00e7\u00e3o\" title=\"Clique aqui.\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Solu\u00e7\u00e3o';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<div class=\"gr1 cb93\"  style=\"width:90%;line-height:40px;\">\nSe [tex]d_1, \\, d_2, \\, \\cdots \\, , \\, d_r[\/tex] s\u00e3o os divisores positivos de um n\u00famero natural [tex]n[\/tex], observe que:<\/p>\n<ul>\n<li>Para cada [tex]d_i[\/tex], [tex]i=1, \\, 2, \\,  \\, \\cdots \\, , \\, r[\/tex], [tex] \\dfrac{n}{d_i}[\/tex] \u00e9 tamb\u00e9m divisor de [tex]n[\/tex].[tex]\\qquad \\qquad (i)[\/tex]<br \/>\nCom efeito, como cada [tex]d_i[\/tex] divide [tex]n[\/tex], ent\u00e3o [tex]n=d_i\\cdot k[\/tex], com [tex]k\\in\\mathbb{N}[\/tex]. Assim [tex]k=\\dfrac{n}{d_i}[\/tex] e, ent\u00e3o,  [tex]\\dfrac{n}{d_i}\\in\\mathbb{N}[\/tex]. Por outro lado, [tex]n=\\dfrac{n}{d_i}\\cdot d_i[\/tex], com [tex]n, \\, \\dfrac{n}{d_i}\\in\\mathbb{N}[\/tex], portanto, [tex] \\dfrac{n}{d_i}[\/tex] \u00e9 divisor de [tex]n[\/tex].<\/li>\n<li>Para [tex]i, \\, j=1, \\, 2, \\,  \\, \\cdots \\, , \\, r[\/tex], se [tex]\\dfrac{n}{d_i}=\\dfrac{n}{d_j}[\/tex], ent\u00e3o, claramente, [tex]d_i=d_j[\/tex].[tex]\\qquad(ii) \\, [\/tex]<\/li>\n<\/ul>\n<p>Por [tex](i) \\, [\/tex] e [tex] \\, (ii)[\/tex], conclu\u00edmos que se [tex]\\left\\{d_1, \\, d_2, \\, \\cdots \\, , \\, d_r\\right\\}[\/tex] \u00e9 o conjunto de todos os divisores de [tex]n[\/tex], ent\u00e3o [tex]\\left\\{\\dfrac{n}{d_1}, \\, \\dfrac{n}{d_2}, \\, \\cdots \\, ,\\dfrac{n}{d_r}\\right\\}[\/tex] tamb\u00e9m o \u00e9. Dessa forma, temos que [tex] d_1+d_2+\\cdots+d_r=\\dfrac{n}{d_1}+\\dfrac{n}{d_2}+\\cdots+\\dfrac{n}{d_r}[\/tex].<br \/>\nSendo [tex]n[\/tex]  perfeito, ent\u00e3o [tex]\\sigma(n)=2n[\/tex], donde<br \/>\n[tex]\\begin{align*} \\qquad<br \/>\n  \\dfrac{1}{d_1} + \\dfrac{1}{d_2} + \\cdots + \\dfrac{1}{d_r}  &#038; = \\dfrac{n}{n} \\left( \\dfrac{1}{d_1} + \\dfrac{1}{d_2} + \\cdots + \\dfrac{1}{d_r} \\right) \\\\ \\\\ &#038; =\\dfrac{1}{n} \\left( \\dfrac{n}{d_1} + \\dfrac{n}{d_2} + \\cdots + \\dfrac{n}{d_r} \\right) \\\\ \\\\ &#038;  =\\dfrac{1}{n} \\left( d_1 + d_2 + \\cdots + d_r \\right) \\\\  \\\\ &#038; =\\dfrac{\\sigma(n)}{n}\\\\ \\\\ &#038; =\\dfrac{2n}{n} = 2.<br \/>\n\\end{align*}[\/tex]\n<\/p><\/div>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb94\"  style=\"width:80%;\">\n<strong>Problema 9:<\/strong> <\/p>\n<p>Seja [tex]d[\/tex] um divisor pr\u00f3prio do n\u00famero natural  [tex]n[\/tex] ( [tex]d[\/tex] um divisor de  [tex]n[\/tex], mas [tex]d \\ne n[\/tex]).<br \/>\nMostre que se [tex]n[\/tex] \u00e9 um n\u00famero perfeito, ent\u00e3o [tex]d[\/tex] n\u00e3o \u00e9  perfeito.<\/p>\n<div style=\"text-align: center;\">\n<input  class=\"bo3 c9 \" style=\" margin-left: auto; margin-right: auto;\" value=\"Dica\" title=\"Clique aqui.\" type=\"button\" onclick=    \"if(this.parentNode.getElementsByTagName('div')[0].style.display != ''){this.parentNode.getElementsByTagName('div')[0].style.display = '';this.value = 'Ocultar';} else{this.parentNode.getElementsByTagName('div')[0].style.display = 'none'; this.value = 'Dica';}\" \/>    <\/p>\n<div  style=\"display: none;\">\n<div class=\"gr1 cb93\" style=\"width:90%;\">\nO resultado anterior garante que<\/p>\n<ul>\n<li>se  [tex]d_1, \\, d_2, \\, \\cdots \\, , \\, d_r[\/tex] s\u00e3o os divisores positivos de um n\u00famero perfeito [tex]m[\/tex], ent\u00e3o [tex] \\, \\dfrac{1}{d_1}+\\dfrac{1}{d_2}+\\cdots+\\dfrac{1}{d_r}=2[\/tex];<\/li>\n<\/ul>\n<p>assim<\/p>\n<ul>\n<li>se [tex]d_1, \\, d_2, \\, \\cdots \\, , \\, d_r[\/tex] s\u00e3o os divisores positivos de um n\u00famero [tex]m[\/tex] e [tex] \\, \\dfrac{1}{d_1}+\\dfrac{1}{d_2}+\\cdots+\\dfrac{1}{d_r}\\ne 2[\/tex], ent\u00e3o [tex]m[\/tex] n\u00e3o \u00e9 perfeito.<\/li>\n<\/ul>\n<p>Portanto, mostre que se [tex]k_1, \\, k_2, \\, \\cdots \\, , \\, k_t[\/tex] s\u00e3o os divisores positivos de [tex]d[\/tex], ent\u00e3o [tex] \\, \\dfrac{1}{k_1}+\\dfrac{1}{k_2}+\\cdots+\\dfrac{1}{k_t}\\ne 2[\/tex].\n<\/p><\/div>\n<\/div>\n<\/div>\n<p style=\"margin:10px;\">\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb94\"  style=\"width:80%;\">\n<strong>Problema 10:<\/strong> <\/p>\n<p>Mostre que todo m\u00faltiplo de um n\u00famero perfeito [tex]n[\/tex] que seja diferente do zero e do pr\u00f3prio [tex]n[\/tex] \u00e9 abundante.<\/p>\n<p style=\"margin:10px;\">\n<\/div>\n<p style=\"margin:20px;\">\n<div class=\"gree cb94\"  style=\"width:80%;\">\n<strong>Problema 11:<\/strong> <\/p>\n<p>Seja [tex]n[\/tex] um n\u00famero perfeito par.<br \/>\nMostre que se [tex]d[\/tex] \u00e9 um divisor de [tex]n[\/tex] tal que [tex]0\\lt d \\lt n[\/tex], ent\u00e3o [tex]d[\/tex] \u00e9 um n\u00famero deficiente.<\/p>\n<p style=\"margin:10px;\">\n<\/div>\n<p style=\"margin:40px;\">\n<hr style=\"height:2px;background-color:#6fa34c;  margin-top: 8px; margin-bottom: 8px;border: 0 none\"\/>\n<hr style=\"height:2px;background-color:#6fa34c;  margin-top: 8px; margin-bottom: 8px;border: 0 none\"\/>\n<p style=\"text-align: right;\"><span style=\"color: #6fa34c; font-size: 20px;\"><strong>Equipe COM &#8211; OBMEP<\/strong><\/span><\/p>\n<\/div>\n<p class=\"fim\"><a title=\"Voltar.\" href=\"http:\/\/clubes.obmep.org.br\/blog\/?p=58445\" target=\"_blank\"><span style=\"color: #6cda00;\"><strong>Voltar para Sala Principal<\/strong><\/span><\/a><\/p>\n<p style=\"margin:30px;\">\n","protected":false},"excerpt":{"rendered":"<p>N\u00fameros perfeitos Problemas Problema 1: (i) Mostrar que 672 \u00e9 um n\u00famero multiperfeito. Qual a sua ordem? (ii) Mostrar que 30240 \u00e9 um n\u00famero multiperfeito de ordem 4. (iii) Mostrar que 523776 \u00e9 um n\u00famero multiperfeito de ordem 3. (iv) Mostrar que 2178540 \u00e9 um n\u00famero multiperfeito de ordem 4. Problema 2: Mostre que um &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/numeros-especiais-numeros-perfeitos\/numeros-especiais-numeros-perfeitos-problemas\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":58445,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-58443","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/58443","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\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=58443"}],"version-history":[{"count":16,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/58443\/revisions"}],"predecessor-version":[{"id":169379,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/58443\/revisions\/169379"}],"up":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/58445"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=58443"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}