{"id":795,"date":"2012-03-08T11:17:10","date_gmt":"2012-03-08T19:17:10","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=795"},"modified":"2020-01-15T10:50:51","modified_gmt":"2020-01-15T18:50:51","slug":"gmat-quant-finding-the-units-digits-of-large-powers","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/","title":{"rendered":"GMAT Quant: Finding the Units Digits of Large Powers"},"content":{"rendered":"<p>Raising to a power is iterated multiplication. Luckily, you can find your units digit with a simple multiplication pattern, even when you&#8217;re working with large powers. (For a refresh of the multiplication rules for unit digits, see our post on <a href=\"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/\">difficult units digits<\/a>.)<\/p>\n<p>See how you do with this question:<\/p>\n<p>What is the units digit of 57<sup>45<\/sup>?<\/p>\n<p>A) 1<\/p>\n<p>B) 3<\/p>\n<p>C) 5<\/p>\n<p>D) 7<\/p>\n<p>E) 9<\/p>\n<p>&nbsp;<\/p>\n<p>To solve this, we\u2019ll begin examining smaller powers and look for a pattern.<\/p>\n<p>57<sup>1<\/sup> = 57\u00a0(the units digit is 7)<\/p>\n<p>57<sup>2<\/sup> = 3,249\u00a0(the units digit is 9)<\/p>\n<p>57<sup>3<\/sup> = 185,193\u00a0(the units digit is 3)<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-decoration: underline;\">Aside<\/span>: Since these powers increase quickly, it\u2019s useful to notice that we need only multiply the units digit each time. For example, the units digit of 57<sup>2<\/sup>\u00a0is the same as the units digit of 7<sup>2<\/sup>. Similarly, the units digit of 57<sup>3<\/sup>\u00a0is the same as the units digit of 7<sup>3<\/sup>.<\/p>\n<p>So, once we know that the units of 57<sup>2<\/sup>\u00a0is 9, we can find the units digit of 57<sup>3<\/sup>\u00a0by multiplying 9 by 7 to get 63. So the units digit of 57<sup>3<\/sup>\u00a0is 3.<\/p>\n<p>To find the units digit of 57<sup>4<\/sup>, we\u2019ll multiply 3 by 7 to get 21. So the units digit of 57<sup>4<\/sup>\u00a0is 1.<\/p>\n<p>When we start listing the various powers, we can see a pattern emerge:<\/p>\n<ul>\n<li>The units digit of 57<sup>1<\/sup>\u00a0is 7<\/li>\n<li>The units digit of 57<sup>2<\/sup>\u00a0is 9<\/li>\n<li>The units digit of 57<sup>3<\/sup>\u00a0is 3<\/li>\n<li>The units digit of 57<sup>4<\/sup>\u00a0is 1<\/li>\n<li>The units digit of 57<sup>5<\/sup>\u00a0is 7<\/li>\n<\/ul>\n<p>At this point, we should recognize that the pattern begins to repeat. The pattern goes: 7-9-3-1-7-9-3-1-7-9-3-1-\u2026<\/p>\n<p>Since the pattern repeats itself every 4 powers, we say that the \u201c<strong>cycle<\/strong>\u201d equals <strong>4<\/strong><\/p>\n<p>Now comes an important observation:<\/p>\n<p>The units digit of 57<sup>1<\/sup>\u00a0is 7<\/p>\n<p>The units digit of 57<sup>2<\/sup>\u00a0is 9<\/p>\n<p>The units digit of 57<sup>3<\/sup>\u00a0is 3<\/p>\n<p>The units digit of 57<sup>4<\/sup>\u00a0is <strong>1<\/strong><\/p>\n<p>The units digit of 57<sup>5<\/sup>\u00a0is 7<\/p>\n<p>The units digit of 57<sup>6<\/sup>\u00a0is 9<\/p>\n<p>The units digit of 57<sup>7<\/sup>\u00a0is 3<\/p>\n<p>The units digit of 57<sup>8<\/sup> is <strong>1<\/strong><\/p>\n<p>The units digit of 57<sup>9<\/sup>\u00a0is 7<\/p>\n<p>The units digit of 57<sup>10<\/sup>\u00a0is 9<\/p>\n<p>The units digit of 57<sup>11<\/sup>\u00a0is 3<\/p>\n<p>The units digit of 57<sup>12<\/sup>\u00a0is <strong>1<\/strong>. . .<strong> <\/strong>etc.<\/p>\n<p>As you can see, since the cycle = <strong>4<\/strong>, the units digit of 57<sup><em>k<\/em><\/sup>\u00a0will be <strong>1<\/strong> <strong>whenever <em>k<\/em> is a multiple of<\/strong> <strong>4<\/strong>.<\/p>\n<p>Now to find the units digit of 57<sup>45<\/sup>, all we need to do is recognize that the units digit of 57<sup>44<\/sup>\u00a0is <strong>1 <\/strong>(since 44 is a multiple of <strong>4<\/strong>).<\/p>\n<p>From here, we\u2019ll just continue with our pattern:<\/p>\n<p>The units digit of 57<sup>44<\/sup>\u00a0is <strong>1<\/strong><\/p>\n<p>The units digit of 57<sup>45<\/sup>\u00a0is 7<\/p>\n<p>The units digit of 57<sup>46<\/sup>\u00a0is 9<\/p>\n<p>The units digit of 57<sup>47<\/sup>\u00a0is 3 . . . etc.<\/p>\n<p>So, the units digit of 57<sup>45<\/sup>\u00a0is 7, which means the answer is D.<\/p>\n<p>&nbsp;<\/p>\n<p>If you\u2019d like to practice, you can answer these two questions:<\/p>\n<ol>\n<li>What is the units digit of 83<sup>75<\/sup>?<\/li>\n<li>What is the units digit of 39<sup>61<\/sup>?<\/li>\n<\/ol>\n<p>(The answers can be found at the very bottom of this post)<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><em>Answers: <\/em><\/p>\n<p><em>1. 7<\/em><\/p>\n<p><em>2. 9<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Raising to a power is iterated multiplication. Luckily, you can find your units digit with a simple multiplication pattern, even when you&#8217;re working with large powers. (For a refresh of the multiplication rules for unit digits, see our post on difficult units digits.) See how you do with this question: What is the units digit [&hellip;]<\/p>\n","protected":false},"author":11,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[112],"tags":[],"ppma_author":[13213],"class_list":["post-795","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>GMAT Quant: Finding the Units Digits of Large Powers - Magoosh Blog \u2014 GMAT\u00ae Exam<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"GMAT Quant: Finding the Units Digits of Large Powers\" \/>\n<meta property=\"og:description\" content=\"Raising to a power is iterated multiplication. Luckily, you can find your units digit with a simple multiplication pattern, even when you&#8217;re working with large powers. (For a refresh of the multiplication rules for unit digits, see our post on difficult units digits.) See how you do with this question: What is the units digit [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/\" \/>\n<meta property=\"og:site_name\" content=\"Magoosh Blog \u2014 GMAT\u00ae Exam\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/MagooshGMAT\/\" \/>\n<meta property=\"article:published_time\" content=\"2012-03-08T19:17:10+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2020-01-15T18:50:51+00:00\" \/>\n<meta name=\"author\" content=\"Brent\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@MagooshGMAT\" \/>\n<meta name=\"twitter:site\" content=\"@MagooshGMAT\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Brent\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/\"},\"author\":{\"name\":\"Brent\",\"@id\":\"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/b2fbd67daad89818fde8859916e23409\"},\"headline\":\"GMAT Quant: Finding the Units Digits of Large Powers\",\"datePublished\":\"2012-03-08T19:17:10+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/\"},\"wordCount\":465,\"commentCount\":49,\"publisher\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/#organization\"},\"articleSection\":[\"GMAT Math\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/\",\"url\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/\",\"name\":\"GMAT Quant: Finding the Units Digits of Large Powers - Magoosh Blog \u2014 GMAT\u00ae Exam\",\"isPartOf\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/#website\"},\"datePublished\":\"2012-03-08T19:17:10+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/magoosh.com\/gmat\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"GMAT Quant: Finding the Units Digits of Large Powers\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/magoosh.com\/gmat\/#website\",\"url\":\"https:\/\/magoosh.com\/gmat\/\",\"name\":\"Magoosh Blog \u2014 GMAT\u00ae Exam\",\"description\":\"Everything you need to know about the GMAT\",\"publisher\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/magoosh.com\/gmat\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/magoosh.com\/gmat\/#organization\",\"name\":\"Magoosh\",\"url\":\"https:\/\/magoosh.com\/gmat\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/magoosh.com\/gmat\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/magoosh.com\/gmat\/files\/2019\/04\/Magoosh-logo-purple-60h.png\",\"contentUrl\":\"https:\/\/magoosh.com\/gmat\/files\/2019\/04\/Magoosh-logo-purple-60h.png\",\"width\":265,\"height\":60,\"caption\":\"Magoosh\"},\"image\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.facebook.com\/MagooshGMAT\/\",\"https:\/\/twitter.com\/MagooshGMAT\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/b2fbd67daad89818fde8859916e23409\",\"name\":\"Brent\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/image\/2c44c3d2b5d811cb2cc2f7b52e98a186\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/10b86d22636e4b773ca09b355ca87734de1952dba2b74eaa4ce2f13249accba4?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/10b86d22636e4b773ca09b355ca87734de1952dba2b74eaa4ce2f13249accba4?s=96&d=mm&r=g\",\"caption\":\"Brent\"},\"description\":\"Brent Hanneson is a master tutor with over 20 years of teaching experience. He developed all the math content for Magoosh Test Prep. Brent plays ice hockey in his free time.\",\"url\":\"https:\/\/magoosh.com\/gmat\/author\/brent\/\"}]}<\/script>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"GMAT Quant: Finding the Units Digits of Large Powers - Magoosh Blog \u2014 GMAT\u00ae Exam","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/","og_locale":"en_US","og_type":"article","og_title":"GMAT Quant: Finding the Units Digits of Large Powers","og_description":"Raising to a power is iterated multiplication. Luckily, you can find your units digit with a simple multiplication pattern, even when you&#8217;re working with large powers. (For a refresh of the multiplication rules for unit digits, see our post on difficult units digits.) See how you do with this question: What is the units digit [&hellip;]","og_url":"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/","og_site_name":"Magoosh Blog \u2014 GMAT\u00ae Exam","article_publisher":"https:\/\/www.facebook.com\/MagooshGMAT\/","article_published_time":"2012-03-08T19:17:10+00:00","article_modified_time":"2020-01-15T18:50:51+00:00","author":"Brent","twitter_card":"summary_large_image","twitter_creator":"@MagooshGMAT","twitter_site":"@MagooshGMAT","twitter_misc":{"Written by":"Brent","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/#article","isPartOf":{"@id":"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/"},"author":{"name":"Brent","@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/b2fbd67daad89818fde8859916e23409"},"headline":"GMAT Quant: Finding the Units Digits of Large Powers","datePublished":"2012-03-08T19:17:10+00:00","mainEntityOfPage":{"@id":"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/"},"wordCount":465,"commentCount":49,"publisher":{"@id":"https:\/\/magoosh.com\/gmat\/#organization"},"articleSection":["GMAT Math"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/","url":"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/","name":"GMAT Quant: Finding the Units Digits of Large Powers - Magoosh Blog \u2014 GMAT\u00ae Exam","isPartOf":{"@id":"https:\/\/magoosh.com\/gmat\/#website"},"datePublished":"2012-03-08T19:17:10+00:00","breadcrumb":{"@id":"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/magoosh.com\/gmat\/gmat-quant-finding-the-units-digits-of-large-powers\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/magoosh.com\/gmat\/"},{"@type":"ListItem","position":2,"name":"GMAT Quant: Finding the Units Digits of Large Powers"}]},{"@type":"WebSite","@id":"https:\/\/magoosh.com\/gmat\/#website","url":"https:\/\/magoosh.com\/gmat\/","name":"Magoosh Blog \u2014 GMAT\u00ae Exam","description":"Everything you need to know about the GMAT","publisher":{"@id":"https:\/\/magoosh.com\/gmat\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/magoosh.com\/gmat\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/magoosh.com\/gmat\/#organization","name":"Magoosh","url":"https:\/\/magoosh.com\/gmat\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/logo\/image\/","url":"https:\/\/magoosh.com\/gmat\/files\/2019\/04\/Magoosh-logo-purple-60h.png","contentUrl":"https:\/\/magoosh.com\/gmat\/files\/2019\/04\/Magoosh-logo-purple-60h.png","width":265,"height":60,"caption":"Magoosh"},"image":{"@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/MagooshGMAT\/","https:\/\/twitter.com\/MagooshGMAT"]},{"@type":"Person","@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/b2fbd67daad89818fde8859916e23409","name":"Brent","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/image\/2c44c3d2b5d811cb2cc2f7b52e98a186","url":"https:\/\/secure.gravatar.com\/avatar\/10b86d22636e4b773ca09b355ca87734de1952dba2b74eaa4ce2f13249accba4?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/10b86d22636e4b773ca09b355ca87734de1952dba2b74eaa4ce2f13249accba4?s=96&d=mm&r=g","caption":"Brent"},"description":"Brent Hanneson is a master tutor with over 20 years of teaching experience. He developed all the math content for Magoosh Test Prep. Brent plays ice hockey in his free time.","url":"https:\/\/magoosh.com\/gmat\/author\/brent\/"}]}},"authors":[{"term_id":13213,"user_id":11,"is_guest":0,"slug":"brent","display_name":"Brent","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/10b86d22636e4b773ca09b355ca87734de1952dba2b74eaa4ce2f13249accba4?s=96&d=mm&r=g","user_url":"","last_name":"Hanneson","first_name":"Brent","description":"Brent Hanneson is a master tutor with over 20 years of teaching experience. He developed all the math content for Magoosh Test Prep. Brent plays ice hockey in his free time."}],"_links":{"self":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/795","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/users\/11"}],"replies":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/comments?post=795"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/795\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/media?parent=795"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/categories?post=795"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/tags?post=795"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/ppma_author?post=795"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}