{"id":65635,"date":"2020-11-27T22:10:18","date_gmt":"2020-11-27T16:40:18","guid":{"rendered":"https:\/\/www.oliveboard.in\/blog\/?p=65635"},"modified":"2021-02-25T09:44:23","modified_gmt":"2021-02-25T04:14:23","slug":"remainder-theorem-and-unit-digit","status":"publish","type":"post","link":"https:\/\/www.oliveboard.in\/blog\/remainder-theorem-and-unit-digit\/","title":{"rendered":"Practice Questions on Remainder Theorem and Unit Digit for SSC"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_77 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of content<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.oliveboard.in\/blog\/remainder-theorem-and-unit-digit\/#Important_concepts_of_Remainder_theorem_and_unit_digit\" >Important concepts of Remainder theorem and unit digit<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.oliveboard.in\/blog\/remainder-theorem-and-unit-digit\/#Remainder_theorem_and_unit_digit_Some_Important_Points_to_Remember\" >Remainder theorem and unit digit: Some Important Points to Remember<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.oliveboard.in\/blog\/remainder-theorem-and-unit-digit\/#Practice_questions_on_remainder_theorem_and_unit_digit\" >Practice questions on remainder theorem and unit digit<\/a><\/li><\/ul><\/nav><\/div>\n<p>Number system is the most fundamental topic in mathematics, one of whose subtopics include remainder theorem and unit digit. In today\u2019s blog we will be providing you with some practice questions on remainder theorem and unit digit for SSC. First of all, let us go through the basic concepts.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Important_concepts_of_Remainder_theorem_and_unit_digit\"><\/span><strong>Important concepts of Remainder theorem and unit digit<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3><strong>Basic remainder formula:<\/strong><\/h3>\n<p>Dividend = Divisor * Quotient + Remainder<\/p>\n<p>If remainder = 0, then it the number is perfectly divisible by divisor.<\/p>\n<p><strong>Note: <\/strong>If we have a negative remainder (which was assumed for the convenience during solving of the problem), the divisor is to be added to get the real remainder.<\/p>\n<h3><strong>Cyclicity:<\/strong><\/h3>\n<p>Cyclicity is an important concept which can be used to solve questions on remainder theorem and unit digit.<\/p>\n<table id=\"tg-BrEUC\" class=\"tg\" style=\"border-collapse: collapse; border-spacing: 0px; height: 90px;\" width=\"684\">\n<thead>\n<tr>\n<th style=\"background-color: #a6a6a6; font-family: Arial, sans-serif; font-size: 14px; font-weight: normal; overflow: hidden; padding: 0px 0px; text-align: left; vertical-align: top; word-break: normal; border: 1px solid black;\"><span style=\"background-color: #a6a6a6;\">\u00a0 <\/span><span style=\"color: black;\">Numbers<\/span><span style=\"background-color: #a6a6a6;\">\u00a0\u00a0\u00a0<\/span><\/th>\n<th style=\"background-color: #a6a6a6; font-family: Arial, sans-serif; font-size: 14px; font-weight: normal; overflow: hidden; padding: 0px; vertical-align: top; word-break: normal; border: 1px solid black; text-align: left;\"><span style=\"color: black;\">Cyclicity<\/span><span style=\"background-color: #a6a6a6;\">\u00a0\u00a0\u00a0<\/span><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"background-color: #d9d9d9; font-family: Arial, sans-serif; font-size: 14px; overflow: hidden; padding: 0px 0px; text-align: left; vertical-align: top; word-break: normal; border: 1px solid black;\"><span style=\"background-color: #d9d9d9;\">\u00a0<\/span><span style=\"color: black;\">0, 1, 5, 6<\/span><span style=\"background-color: #d9d9d9;\">\u00a0\u00a0\u00a0<\/span><\/td>\n<td style=\"font-family: Arial, sans-serif; font-size: 14px; overflow: hidden; padding: 0px 0px; text-align: left; vertical-align: top; word-break: normal; border: 1px solid black;\">\u00a0<span style=\"color: black;\">1<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"font-family: Arial, sans-serif; font-size: 14px; overflow: hidden; padding: 0px 0px; text-align: left; vertical-align: top; word-break: normal; border: 1px solid black;\">\u00a0 <span style=\"color: black;\">2, 3, 7, 8<\/span><\/td>\n<td style=\"background-color: #d9d9d9; font-family: Arial, sans-serif; font-size: 14px; overflow: hidden; padding: 0px 0px; text-align: left; vertical-align: top; word-break: normal; border: 1px solid black;\"><span style=\"background-color: #d9d9d9;\">\u00a0 <\/span><span style=\"color: black;\">4<\/span><span style=\"background-color: #d9d9d9;\">\u00a0\u00a0\u00a0<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"background-color: #d9d9d9; font-family: Arial, sans-serif; font-size: 14px; overflow: hidden; padding: 0px 0px; text-align: left; vertical-align: top; word-break: normal; border: 1px solid black;\"><span style=\"background-color: #d9d9d9;\">\u00a0 <\/span><span style=\"color: black;\">4, 9<\/span><span style=\"background-color: #d9d9d9;\">\u00a0\u00a0\u00a0<\/span><\/td>\n<td style=\"font-family: Arial, sans-serif; font-size: 14px; overflow: hidden; padding: 0px 0px; text-align: left; vertical-align: top; word-break: normal; border: 1px solid black;\">\u00a0 <span style=\"color: black;\">2<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a href=\"https:\/\/www.oliveboard.in\/ssc-cgl\/?ref=content-mm\"><strong>Take a Free SSC CGL Tier 2 Mock Test for Quant<\/strong><\/a><\/p>\n<h3><strong>Euler\u2019s Remainder theorem:<\/strong><\/h3>\n<p><strong>Euler\u2019s Remainder theorem:<\/strong> For co-prime numbers M and N, Remainder [M<sup>E(N)\u00a0<\/sup>\/ N] = 1, where E(N) is Euler number of N .<\/p>\n<p>If N= a<sup>n<\/sup> * b<sup>s<\/sup> * c<sup>m<\/sup> * d<sup>u<\/sup> * f <sup>r<\/sup> such that a, b, c, d, f are prime numbers<\/p>\n<p>Euler\u2019s number of N = N (1 \u2013 1\/a)*(1 \u2013 1\/b)*(1 \u2013 1\/c)*(1 \u2013 1\/d)* (1 \u2013 1\/f)<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Remainder_theorem_and_unit_digit_Some_Important_Points_to_Remember\"><\/span><strong>Remainder theorem and unit digit<\/strong><strong>: Some Important Points to Remember<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>(a<sup>n\u00a0<\/sup>+ b<sup>n<\/sup>) is divisible by (a + b), when n is odd.<\/li>\n<li>(a<sup>n\u00a0<\/sup>&#8211; b<sup>n<\/sup>) is divisible by (a + b), when n is even.<\/li>\n<li>(a<sup>n<\/sup>\u00a0&#8211; b<sup>n<\/sup>) is always divisible by (a &#8211; b), for every n.<\/li>\n<li>Remainder (a*b\/c) = Remainder ((Remainder(a\/c) * Remainder(b\/c))\/c).<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"Practice_questions_on_remainder_theorem_and_unit_digit\"><\/span><strong>Practice questions <\/strong><strong>on remainder theorem and unit digit<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>1) \u00a0What is the unit digit of 1! + 2! + 3! + \u2026\u2026+ 88! + 89!?<\/p>\n<p>(a) 5<\/p>\n<p>(b) 3<\/p>\n<p>(c) 1<\/p>\n<p>(d) 8<\/p>\n<p>Answer key: b<\/p>\n<p>Solution:<\/p>\n<p>1! =1<\/p>\n<p>2! = 2<\/p>\n<p>3! = 6<\/p>\n<p>4! = 24<\/p>\n<p>5! = 120<\/p>\n<p>6! = 620&#8230;..<\/p>\n<p>The unit digit of factorial of any number greater than 4 is 0. Hence,<\/p>\n<p>Unit digit of 1! + 2! + 3! + \u2026\u2026+ 88! + 89! = Unit digit of 1 + 2 + 6 + 4 + 0 + 0 + &#8230;&#8230;+ 0<\/p>\n<p>= Unit digit of 13 = 3<\/p>\n<p><a href=\"https:\/\/www.oliveboard.in\/ssc-cgl\/?ref=content-mm\"><strong>Take a Free SSC CGL Tier 2 Mock Test for Quant<\/strong><\/a><\/p>\n<p>2) If 3 divided the integer n, the remainder is 2. Then, what will be the remainder when 7n is divided by 3<\/p>\n<p>(a) 3<\/p>\n<p>(b) 2<\/p>\n<p>(c) 6<\/p>\n<p>(d) 4<\/p>\n<p>Answer key: b<\/p>\n<p>Solution:<\/p>\n<p>Remainder (7n\/3) = Remainder ((Remainder (7\/3) * Remainder(n\/3))\/3) = Remainder (1*2\/3) = 2<\/p>\n<p><a href=\"https:\/\/www.oliveboard.in\/free-ebook-pdf-download\/?ref=content-mm\"><strong>Download FREE e-books on various topics of Quant here<\/strong><\/a><\/p>\n<p>3) What is the remainder when 1294*1298 is divided by 16.<\/p>\n<p>(a) 14<\/p>\n<p>(b) 11<\/p>\n<p>(c) 12<\/p>\n<p>(d) 10<\/p>\n<p>Answer key: c<\/p>\n<p>Solution:<\/p>\n<p>16*81 =1296<\/p>\n<p>Remainder (1294\/16) = -2 {1294 = 16*81 &#8211; 2}<\/p>\n<p>Remainder (1298\/16) = 2 {1298 = 16*81 + 2}<\/p>\n<p>Remainder (1294*1298\/16) = -2*2 = -4 = -4+16 = 12<\/p>\n<p><a href=\"https:\/\/www.oliveboard.in\/ssc-cgl\/?ref=content-mm\"><strong>Take a Free SSC CGL Tier 2 Mock Test for Quant<\/strong><\/a><\/p>\n<p>4) 3<sup>10 <\/sup>+ 5<sup>10<\/sup> is divisible by<\/p>\n<p>(a) 34<\/p>\n<p>(b) 26<\/p>\n<p>(c) 8<\/p>\n<p>(d) 20<\/p>\n<p>Answer key: a<\/p>\n<p>Solution:<\/p>\n<p>3<sup>10 <\/sup>+ 5<sup>10<\/sup> = 9<sup>5<\/sup> + 25<sup>5<\/sup><\/p>\n<p>We know, (a<sup>n\u00a0<\/sup>+ b<sup>n<\/sup>) is divisible by (a + b), when n is odd.<\/p>\n<p>So,<\/p>\n<p>3<sup>10 <\/sup>+ 5<sup>10<\/sup> is divisible by 9 + 25 = 34<\/p>\n<p><a href=\"https:\/\/www.oliveboard.in\/free-ebook-pdf-download\/?ref=content-mm\"><strong>Download FREE e-books on various topics of Quant here<\/strong><\/a><\/p>\n<p>5) What is remainder obtained if 455<sup>18<\/sup> is divided by 19<\/p>\n<p>(a) 0<\/p>\n<p>(b) 3<\/p>\n<p>(c) 4<\/p>\n<p>(d) 1<\/p>\n<p>Answer key: d<\/p>\n<p>Solution:<\/p>\n<p>Since 19 is prime, Euler\u2019s number of 19 = 19 (1 \u2013 1\/19) = 18<\/p>\n<p>Hence, by Euler\u2019s remainder theorem, the remainder = 1<\/p>\n<p><a href=\"https:\/\/www.oliveboard.in\/ssc-cgl\/?ref=content-mm\"><strong>Take a Free SSC CGL Tier 2 Mock Test for Quant<\/strong><\/a><\/p>\n<p>6) What is the remainder of 1<sup>5<\/sup>+2<sup>5<\/sup>+ 3<sup>5<\/sup> + 4<sup>5 <\/sup>+ 5<sup>5 <\/sup>+ 6<sup>5<\/sup>+7<sup>5<\/sup>+&#8230;..+ 50<sup>5<\/sup> when divided by 5<\/p>\n<p>(a) 3<\/p>\n<p>(b) 4<\/p>\n<p>(c) 2<\/p>\n<p>(d) 0<\/p>\n<p>Answer key: d<\/p>\n<p>Solution:<\/p>\n<p>When the power \u20185\u2019 is divided by cyclicity of the numbers 0, 1, 5 and 6, the remainder = 1<\/p>\n<p>When the power \u20185\u2019 is divided by cyclicity of the numbers 2, 3, 7 and 8, the remainder = 1<\/p>\n<p>When the power \u20185\u2019 is divided by cyclicity of the numbers 4 and 9, the remainder = 1<\/p>\n<p>1<sup>5<\/sup>+2<sup>5<\/sup>+ 3<sup>5<\/sup> + 4<sup>5 <\/sup>+ 5<sup>5 <\/sup>+ 6<sup>5<\/sup>+7<sup>5<\/sup>+&#8230;..+ 50<sup>5<\/sup> =1<sup>1<\/sup>+2<sup>1<\/sup>+ 3<sup>1<\/sup> + 4<sup>1 <\/sup>+ 5<sup>1 <\/sup>+ 6<sup>1<\/sup>+7<sup>1<\/sup>+&#8230;..+ 50<sup>1<\/sup> = Sum of first 50 natural numbers = 50*51\/2 = 1275, divisible by 5 and hence remainder = 0<\/p>\n<p><a href=\"https:\/\/www.oliveboard.in\/ssc-cgl\/?ref=content-mm\"><strong>Take a Free SSC CGL Tier 2 Mock Test for Quant<\/strong><\/a><\/p>\n<p>7) What is the unit digit of 287*586*878<\/p>\n<p>(a) 6<\/p>\n<p>(b) 9<\/p>\n<p>(c) 2<\/p>\n<p>(d) 4<\/p>\n<p>Answer key: a<\/p>\n<p>Solution:<\/p>\n<p>Unit digit of 287*586*878 = Unit digit of 7*6*8 = Unit digit of 336 = 6<\/p>\n<p>Practice more to increase your speed of solving. All the best for your examination!<\/p>\n<p><a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.oliveboard.prep\"><strong>Study at your own convenience anywhere. Download the Oliveboard app now!<\/strong><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Number system is the most fundamental topic in mathematics, one of whose subtopics include remainder theorem and unit digit. In<\/p>\n","protected":false},"author":2,"featured_media":65636,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,2151,5217,10231,63],"tags":[],"class_list":["post-65635","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-articles","category-quiz_test","category-ssc-prep","category-ssc-railways","category-ssc-exams","generate-columns","tablet-grid-50","mobile-grid-100","grid-parent","grid-50"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.6 (Yoast SEO v26.6) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Practice questions on remainder theorem and unit digit for SSC<\/title>\n<meta name=\"description\" content=\"In this blog, some practice questions on remainder theorem and unit digit for SSC are given along with the basic concepts.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.oliveboard.in\/blog\/remainder-theorem-and-unit-digit\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Practice Questions on Remainder Theorem and Unit Digit for SSC\" \/>\n<meta property=\"og:description\" content=\"Number system is the most fundamental topic in mathematics, one of whose subtopics include remainder theorem and unit digit. 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