{"id":331,"date":"2019-06-26T23:37:55","date_gmt":"2019-06-26T23:37:55","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=331"},"modified":"2019-12-20T22:42:23","modified_gmt":"2019-12-20T22:42:23","slug":"repunit-fun","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/repunit-fun\/","title":{"rendered":"Repunit Fun"},"content":{"rendered":"\n<p>A&nbsp;<em>repunit<\/em>&nbsp;is a whole number consisting of only 1&#8217;s, such as 1, 111, or 111111111. (It&#8217;s like a&nbsp;<em>rep<\/em>eating&nbsp;<em>unit<\/em>.)<\/p>\n\n\n\n<p>These numbers have some fun properties. For instance, many have already noted that the square of a repunit exhibits a nice pattern:<\/p>\n\n\n\n<p style=\"text-align:center\">111*111 = 12321,<br>1111*1111 = 1234321,<br>11111*11111 = 123454321,<\/p>\n\n\n\n<p>though this pattern is somewhat messed up by carrying digits when the repunit size is bigger than 9.<\/p>\n\n\n\n<p>Here&#8217;s another less well known pattern:<\/p>\n\n\n\n<p style=\"text-align:center\">222+(333)<sup>2<\/sup>\u00a0= 111111,<br>2222+(3333)<sup>2<\/sup>\u00a0= 11111111,<br>22222+(33333)<sup>2<\/sup>\u00a0= 1111111111.<\/p>\n\n\n\n<p>You might see if you can figure out why.<\/p>\n\n\n\n<p><strong>Presentation&nbsp;Suggestions:<\/strong><br>Can you find other cool properties of repunits?<\/p>\n\n\n\n<p><strong>The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong><br>This property and many such properties can be proved by writing an n-digit repunit as:<\/p>\n\n\n\n<p style=\"text-align:center\">(10<sup>n<\/sup>\u00a0&#8211; 1 )\/9<\/p>\n\n\n\n<p>which follows from noting the repunit is a sum of a geometric series with ratio 10. For n=4, this is just the identity 1111 = 9999\/9.<\/p>\n\n\n\n<p>You might also check out the Fun Facts\u00a0Multiplication by 11\u00a0and\u00a0Multiplication by 111.<\/p>\n\n\n\n<p><strong>How to Cite this Page:<\/strong>&nbsp;<br>Su, Francis E., et al. &#8220;Repunit Fun.&#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>Francis Su&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A&nbsp;repunit&nbsp;is a whole number consisting of only 1&#8217;s, such as 1, 111, or 111111111. (It&#8217;s like a&nbsp;repeating&nbsp;unit.) These numbers have&#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":[3,72,10,12],"class_list":["post-331","page","type-page","status-publish","hentry","tag-easy","tag-geometric-series","tag-numtheory","tag-other"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/331","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=331"}],"version-history":[{"count":4,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/331\/revisions"}],"predecessor-version":[{"id":1607,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/331\/revisions\/1607"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=331"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=331"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}