{"id":1121,"date":"2011-01-15T08:48:04","date_gmt":"2011-01-15T07:48:04","guid":{"rendered":"http:\/\/www.fiveoclock.de\/?p=1121"},"modified":"2011-01-18T21:30:48","modified_gmt":"2011-01-18T20:30:48","slug":"5-fach","status":"publish","type":"post","link":"https:\/\/www.fiveoclock.de\/?p=1121","title":{"rendered":"5-fach"},"content":{"rendered":"<p>Hier mal wieder ein R\u00e4tsel:<\/p>\n<p>gesucht wird die kleinste Zahl, die auf die 7 endet und sich verf\u00fcnffacht, wenn man die 7 als letzte Ziffer entfernt und als erste Ziffer an den Anfag der Zahl stellt.<\/p>\n<p><a class=\"spoiler_link_show\" href=\"javascript:void(0)\" onclick=\"wpSpoilerToggle(document.getElementById('id1043369006'), this, 'Hinweis', 'Verstecken')\">Hinweis<\/a>\n<div class=\"spoiler_div\" id=\"id1043369006\" style=\"display:none\"><\/p>\n<p>Es hilft sich zu \u00fcberlegen, welche Zahl vor der 7 stehen muss.<\/p>\n<p><\/div>\n<\/p>\n<p><a class=\"spoiler_link_show\" href=\"javascript:void(0)\" onclick=\"wpSpoilerToggle(document.getElementById('id744382275'), this, 'L\u00f6sung', 'Ausblenden')\">L\u00f6sung<\/a>\n<div class=\"spoiler_div\" id=\"id744382275\" style=\"display:none\"><\/p>\n<p>Die gesuchte Zahl endet auf 7, hat also die Form:<\/p>\n<p><span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=.........z_%7B1%7D7&#038;fg=00000&#038;s=0\" title=\".........z_{1}7\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\".........z_{1}7\" \/><\/span><\/p>\n<p>Wobei <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=z_1&#038;fg=00000&#038;s=0\" title=\"z_1\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"z_1\" \/><\/span> die noch unbekannte vorletzte Ziffer darstellt.<\/p>\n<p>Nach der Umstellung sieht sie so aus:<\/p>\n<p><span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=7.........z_%7B1%7D&#038;fg=00000&#038;s=0\" title=\"7.........z_{1}\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"7.........z_{1}\" \/><\/span> und weiterhin soll ja gelten:<\/p>\n<p><span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=........z_%7B1%7D7%20%5Ctimes%205%20%3D%207.........z_%7B1%7D&#038;fg=00000&#038;s=0\" title=\"........z_{1}7 \\times 5 = 7.........z_{1}\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"........z_{1}7 \\times 5 = 7.........z_{1}\" \/><\/span><\/p>\n<p>Da <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=5%5Ctimes7%20%3D%2035&#038;fg=00000&#038;s=0\" title=\"5\\times7 = 35\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"5\\times7 = 35\" \/><\/span>\u00a0 ist, mu\u00df <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=z_%7B1%7D&#038;fg=00000&#038;s=0\" title=\"z_{1}\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"z_{1}\" \/><\/span> gleich 5 sein. Gleizeitg ensteht aber auch ein \u00dcbertrag von 3 auf die n\u00e4chste Stelle. Man erh\u00e4lt so:<\/p>\n<p><span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=........%283%2Bz_%7B2%7D%2957%20%5Ctimes%205%20%3D%207.........%283%2Bz_%7B2%7D%295&#038;fg=00000&#038;s=0\" title=\"........(3+z_{2})57 \\times 5 = 7.........(3+z_{2})5\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"........(3+z_{2})57 \\times 5 = 7.........(3+z_{2})5\" \/><\/span><\/p>\n<p>Nun bestimmt man <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=z_%7B2%7D%20%3D%205%20%5Ctimes%205%20%3D%2025&#038;fg=00000&#038;s=0\" title=\"z_{2} = 5 \\times 5 = 25\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"z_{2} = 5 \\times 5 = 25\" \/><\/span>. So ergibt sich als n\u00e4chste Ziffer die 8 (durch den \u00dcbertrag von 3) und man erh\u00e4lt einen neuen \u00dcbertrag von 2:<\/p>\n<p><span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=........%282%2Bz_%7B3%7D%29857%20%5Ctimes%205%20%3D%207.........%282%2Bz_%7B3%7D%2985&#038;fg=00000&#038;s=0\" title=\"........(2+z_{3})857 \\times 5 = 7.........(2+z_{3})85\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"........(2+z_{3})857 \\times 5 = 7.........(2+z_{3})85\" \/><\/span><\/p>\n<p>Durch wiederholte Anwendung:<\/p>\n<p><span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=5%20%5Ctimes%208%20%2B2%20%5Cto%20........%284%2Bz_%7B4%7D%292857%20%5Ctimes%205%20%3D%207.........%284%2Bz_%7B4%7D%29285&#038;fg=00000&#038;s=0\" title=\"5 \\times 8 +2 \\to ........(4+z_{4})2857 \\times 5 = 7.........(4+z_{4})285\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"5 \\times 8 +2 \\to ........(4+z_{4})2857 \\times 5 = 7.........(4+z_{4})285\" \/><\/span><\/p>\n<p><span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=5%20%5Ctimes%202%20%2B%204%20%5Cto%20........%281%2Bz_%7B5%7D%2942857%20%5Ctimes%205%20%3D%207.........%281%2Bz_%7B5%7D%294285&#038;fg=00000&#038;s=0\" title=\"5 \\times 2 + 4 \\to ........(1+z_{5})42857 \\times 5 = 7.........(1+z_{5})4285\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"5 \\times 2 + 4 \\to ........(1+z_{5})42857 \\times 5 = 7.........(1+z_{5})4285\" \/><\/span><\/p>\n<p><span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=5%20%5Ctimes%204%20%2B%201%20%5Cto%20........%282%2Bz_%7B6%7D%29142857%20%5Ctimes%205%20%3D%207.........%282%2Bz_%7B6%7D%2914285&#038;fg=00000&#038;s=0\" title=\"5 \\times 4 + 1 \\to ........(2+z_{6})142857 \\times 5 = 7.........(2+z_{6})14285\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"5 \\times 4 + 1 \\to ........(2+z_{6})142857 \\times 5 = 7.........(2+z_{6})14285\" \/><\/span><\/p>\n<p>Und letzlich ergibt sich mit <span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=5%20%5Ctimes%201%20%2B%202&#038;fg=00000&#038;s=0\" title=\"5 \\times 1 + 2\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"5 \\times 1 + 2\" \/><\/span> als n\u00e4chste Ziffer die 7 und wir haben die L\u00f6sung:<\/p>\n<p><span class=\"tex\"><img decoding=\"async\" src=\"\/tex?x=142857%20%5Ctimes%205%20%3D%20714285&#038;fg=00000&#038;s=0\" title=\"142857 \\times 5 = 714285\" style=\"vertical-align:-20%;\" class=\"tex\" alt=\"142857 \\times 5 = 714285\" \/><\/span><\/p>\n<p><\/div>\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Hier mal wieder ein R\u00e4tsel: gesucht wird die kleinste Zahl, die auf die 7 endet und sich verf\u00fcnffacht, wenn man die 7 als letzte Ziffer entfernt und als erste Ziffer an den Anfag der Zahl stellt. Hinweis Es hilft sich<span class=\"ellipsis\">&hellip;<\/span><\/p>\n<div class=\"read-more\"><a  href=\"https:\/\/www.fiveoclock.de\/?p=1121\">Read more <span class=\"screen-reader-text\">5-fach<\/span><span class=\"meta-nav\"> &#8250;<\/span><\/a><\/div>\n<p><!-- end of .read-more --><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[3],"tags":[],"_links":{"self":[{"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=\/wp\/v2\/posts\/1121"}],"collection":[{"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1121"}],"version-history":[{"count":30,"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=\/wp\/v2\/posts\/1121\/revisions"}],"predecessor-version":[{"id":1150,"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=\/wp\/v2\/posts\/1121\/revisions\/1150"}],"wp:attachment":[{"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1121"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1121"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fiveoclock.de\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1121"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}