{"id":311,"date":"2019-06-26T23:32:41","date_gmt":"2019-06-26T23:32:41","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=311"},"modified":"2019-12-20T22:39:46","modified_gmt":"2019-12-20T22:39:46","slug":"repeating-digits","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/repeating-digits\/","title":{"rendered":"Repeating Digits"},"content":{"rendered":"\n<p>You already know that the\u00a0decimal expansion\u00a0of a rational number eventually repeats or terminates (which can be viewed as a repeating 0).<\/p>\n\n\n\n<p>But I tell you something that perhaps you did not know: if the denominator of that rational number is not divisible by 3, then the&nbsp;<em>repeating part of its decimal expansion is an integer divisible by nine<\/em>!<\/p>\n\n\n\n<p>Example:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>1\/7 = .142857142857&#8230; has repeating part 142857. This is divisible by 9.<\/li><li>41\/55 = .7454545&#8230; has repeating part 45. This is divisible by 9.<\/li><\/ul>\n\n\n\n<p><strong>The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong><br>This rather curious fact can be shown easily. If the rational X is purely repeating of period P and repeating part R, then<\/p>\n\n\n\n<p style=\"text-align:center\">R = 10<sup>P<\/sup>\u00a0X &#8211; X = (10<sup>P<\/sup>-1) X = (10<sup>P<\/sup>-1) (m\/n).<\/p>\n\n\n\n<p>Thus R*n = (10<sup>P<\/sup>-1)*m is an integer. Since (10<sup>P<\/sup>-1) is divisible by 9, if n is not divisible by 3, then R must be. If you like these fun deductions, you may enjoy a course in\u00a0number theory!<\/p>\n\n\n\n<p><strong>How to Cite this Page:<\/strong>&nbsp;<br>Su, Francis E., et al. &#8220;Repeating Digits.&#8221;&nbsp;<em>Math Fun Facts<\/em>. &lt;http:\/\/www.math.hmc.edu\/funfacts&gt;.<\/p>\n\n\n\n<p><strong>Fun Fact suggested by:<\/strong><br>Arthur Benjamin <\/p>\n","protected":false},"excerpt":{"rendered":"<p>You already know that the\u00a0decimal expansion\u00a0of a rational number eventually repeats or terminates (which can be viewed as a repeating&#46;&#46;&#46;<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[3,10],"class_list":["post-311","page","type-page","status-publish","hentry","tag-easy","tag-numtheory"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/311","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=311"}],"version-history":[{"count":4,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/311\/revisions"}],"predecessor-version":[{"id":1606,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/311\/revisions\/1606"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=311"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=311"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}