{"id":758,"date":"2020-08-08T23:20:59","date_gmt":"2020-08-08T23:20:59","guid":{"rendered":"http:\/\/localhost\/?page_id=758"},"modified":"2020-08-08T23:20:59","modified_gmt":"2020-08-08T23:20:59","slug":"exercicios-2","status":"publish","type":"page","link":"http:\/\/localhost\/index.php\/ensino\/algebra-a\/exercicios-2\/","title":{"rendered":"Exerc\u00edcios 2"},"content":{"rendered":"

$\\newcommand{\\Z}{\\mathbb Z}$1. Sejam $a,\\ b,\\ c\\in\\Z$. Demonstre as seguintes afirma\u00e7\u00f5es ou d\u00ea contraexemplo:<\/p>\n

    \n
  1. se $ac\\mid bc$, ent\u00e3o $a\\mid b$;<\/li>\n
  2. se $a\\mid b$ e $a\\mid c$, ent\u00e3o $a\\mid (b-c)$;<\/li>\n
  3. se $c\\mid (a+b)$, ent\u00e3o $c\\mid a$ ou $c\\mid b$;<\/li>\n
  4. se $a\\mid b$, ent\u00e3o $a\\mid xb$ para todo $x\\in\\Z$.<\/li>\n<\/ol>\n

    2. Sejam $a,\\ b,\\ n\\in \\Z$ tais que $n\\geq 2$. Mostre que as seguintes afirma\u00e7\u00f5es s\u00e3o equivalentes:<\/p>\n

      \n
    1. $n\\mid (a-b)$;<\/li>\n
    2. os restos de $a$ e $b$, quando divididos por $n$, s\u00e3o iguais.<\/li>\n<\/ol>\n

      3. Sejam $a,b,c\\in\\Z$. Mostre que<\/p>\n

        \n
      1. $\\mbox{mdc}(ac,bc)=c\\cdot \\mbox{mdc}(a,b)$.<\/li>\n
      2. $\\mbox{mdc}(a\/d,b\/d)=1$ onde $d=\\mbox{mdc}(a,b)$.<\/li>\n<\/ol>\n

        4. Sejam $a, b\\in \\Z$ tais que existem $u,\\ v\\in\\Z$ com $ua+vb=1$. Mostre que $\\mbox{mdc}(a,b)=1$.<\/p>\n

        5. Sejam $a,\\ b,\\ u,\\ v\\in\\Z$ e $d\\in\\mathbb N$ tais que $d=ua+vb$. \u00c9 verdade que
        \n$d=\\mbox{mdc}(a,b)$? Justifique.<\/p>\n

        6. Mostre que dois n\u00fameros naturais consecutivos s\u00e3o primos entre si (ou seja, seu mdc \u00e9 1).<\/p>\n

        7. Mostre para todo $k\\in\\Z$ que $\\mbox{mdc}(4k+3,5k+4)=1$.<\/p>\n","protected":false},"excerpt":{"rendered":"

        $\\newcommand{\\Z}{\\mathbb Z}$1. Sejam $a,\\ b,\\ c\\in\\Z$. Demonstre as seguintes afirma\u00e7\u00f5es ou d\u00ea contraexemplo: se $ac\\mid bc$, ent\u00e3o $a\\mid b$; se $a\\mid b$ e $a\\mid c$, ent\u00e3o $a\\mid (b-c)$; se $c\\mid (a+b)$, ent\u00e3o $c\\mid a$ ou $c\\mid b$; se $a\\mid b$, ent\u00e3o $a\\mid xb$ para todo $x\\in\\Z$. 2. Sejam $a,\\ b,\\ n\\in \\Z$ tais que $n\\geq … Continue reading Exerc\u00edcios 2<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":706,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"_links":{"self":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/758"}],"collection":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/comments?post=758"}],"version-history":[{"count":5,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/758\/revisions"}],"predecessor-version":[{"id":763,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/758\/revisions\/763"}],"up":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/706"}],"wp:attachment":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/media?parent=758"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}