{"id":14702,"date":"2017-09-21T17:13:48","date_gmt":"2017-09-21T17:13:48","guid":{"rendered":"https:\/\/www.manhattanprep.com\/gmat\/?p=14702"},"modified":"2019-09-05T15:51:21","modified_gmt":"2019-09-05T15:51:21","slug":"remainder-cycle-gmat","status":"publish","type":"post","link":"https:\/\/www.manhattanprep.com\/gmat\/blog\/remainder-cycle-gmat\/","title":{"rendered":"The Remainder Cycle"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-14769\" src=\"https:\/\/www.manhattanprep.com\/gmat\/wp-content\/uploads\/sites\/18\/2017\/09\/the-remainder-cycle-reed-arnold.png\" alt=\"Manhattan Prep GMAT Blog - The Remainder Cycle by Reed Arnold\" width=\"1200\" height=\"628\" srcset=\"https:\/\/www.manhattanprep.com\/gmat\/wp-content\/uploads\/sites\/18\/2017\/09\/the-remainder-cycle-reed-arnold.png 1200w, https:\/\/www.manhattanprep.com\/gmat\/wp-content\/uploads\/sites\/18\/2017\/09\/the-remainder-cycle-reed-arnold-300x157.png 300w, https:\/\/www.manhattanprep.com\/gmat\/wp-content\/uploads\/sites\/18\/2017\/09\/the-remainder-cycle-reed-arnold-768x402.png 768w, https:\/\/www.manhattanprep.com\/gmat\/wp-content\/uploads\/sites\/18\/2017\/09\/the-remainder-cycle-reed-arnold-1024x536.png 1024w\" sizes=\"auto, (max-width: 1200px) 100vw, 1200px\" \/><\/p>\n<p><b><i>Guess what? You can attend the first session of any of our online or in-person GMAT courses absolutely free\u2014we\u2019re not kidding! <\/i><\/b><a id=\"bloglink\" href=\"https:\/\/www.manhattanprep.com\/gmat\/classes\/\" target=\"_blank\" rel=\"noopener\"><b><i>Check out our upcoming courses here<\/i><\/b><\/a><b><i>.<\/i><\/b><\/p>\n<hr \/>\n<p><b><i><\/i><\/b><span style=\"font-weight: 400;\">One common complaint I hear from my students is that they \u2018haven\u2019t done math like this since high school.\u2019 And they\u2019re pretty much right: the concepts in the Quant section are by and large wrapped up by Algebra II. But for some subjects, my students drastically underestimate how long it has been since they\u2019ve thought about them. One such subject: remainders on the GMAT.<\/span><!--more--><\/p>\n<p><span style=\"font-weight: 400;\">The last time you probably thought about remainders was early middle school. When you learned long division, you started with things that divided perfectly, and then you learned \u201c25 \u00f7\u00a07 is 3 with 4 left over, so we say it is 3 remainder 4.\u201d But then pretty quickly, you learned about fractions and decimals, so you just said \u201c25 \u00f7\u00a07 is 3 and 4\/7.\u201d Eventually, you just started leaving it as &#8217;25\/7&#8242; and didn\u2019t even worry about it anymore. Remainders, like Destiny\u2019s Child<\/span> <span style=\"font-weight: 400;\">and the purest joys of childhood, faded into distant memory.<\/span><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/giphy.com\/embed\/TT7JW4Qm7uaNW\" width=\"480\" height=\"360\" frameBorder=\"0\" class=\"giphy-embed\" allowFullScreen><\/iframe><\/p>\n<p><a href=\"https:\/\/giphy.com\/gifs\/the-rock-TT7JW4Qm7uaNW\"><\/a><\/p>\n<p><span style=\"font-weight: 400;\">But now they\u2019ve come screaming back into relevance in ways you\u2019ve never thought about them before. The GMAT asks about remainders for a simple reason: they are patterned. Consider this: when dividing any integer by 7, what possible remainders are there? Test a few numbers and see. Could you ever get remainder 8? <\/span><\/p>\n<p><span style=\"font-weight: 400;\">You might realize that the only possible remainders are 1, 2, 3, 4, 5, and 6, but don\u2019t forget you can also get remainder 0\u2014for those numbers that are actually multiples of 7. Do you notice anything about the order in which those remainders come? <\/span><\/p>\n<p><span style=\"font-weight: 400;\">Remainders come in a cycle. When dividing by 7: 8 has remainder 1, 9 has remainder 2, 10 has remainder 3, 11 has remainder 4, 12 has remainder 5, 13 has remainder 6, 14 has remainder 0, 15 has remainder 1, 16 has remainder 2, 17 has remainder 3\u2026 etc. etc. etc., onward and forever. That is, when dividing by 7, the remainders cycle from 0-6 continuously. This is a general rule:<\/span><\/p>\n<p style=\"padding-left: 30px;\"><span style=\"font-weight: 400;\">When dividing integers by integer <\/span><i><span style=\"font-weight: 400;\">n<\/span><\/i><span style=\"font-weight: 400;\">, there are <\/span><i><span style=\"font-weight: 400;\">n<\/span><\/i><span style=\"font-weight: 400;\"> possible remainders, 0 through (<\/span><i><span style=\"font-weight: 400;\">n<\/span><\/i><span style=\"font-weight: 400;\">-1), and these remainders cycle in an infinite loop.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">I call this the \u2018remainder cycle.\u2019 Take a look at the table below to see it in practice for a few smaller numbers. Note that you can say something like \u201c2 \u00f7\u00a09 is 0 remainder 2,\u201d so it actually is said to have a remainder 2.<\/span><\/p>\n<h4><b>Remainder Cycles When Dividing by Integers<\/b><\/h4>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-14703\" src=\"https:\/\/www.manhattanprep.com\/gmat\/wp-content\/uploads\/sites\/18\/2017\/09\/remainder-table.png\" alt=\"Manhattan Prep GMAT Blog - The Remainder Cycle by Reed Arnold\" width=\"1858\" height=\"1046\" srcset=\"https:\/\/www.manhattanprep.com\/gmat\/wp-content\/uploads\/sites\/18\/2017\/09\/remainder-table.png 1858w, https:\/\/www.manhattanprep.com\/gmat\/wp-content\/uploads\/sites\/18\/2017\/09\/remainder-table-300x169.png 300w, https:\/\/www.manhattanprep.com\/gmat\/wp-content\/uploads\/sites\/18\/2017\/09\/remainder-table-768x432.png 768w, https:\/\/www.manhattanprep.com\/gmat\/wp-content\/uploads\/sites\/18\/2017\/09\/remainder-table-1024x576.png 1024w\" sizes=\"auto, (max-width: 1858px) 100vw, 1858px\" \/><\/p>\n<p><span style=\"font-weight: 400;\">Do you see that when you are dividing by <\/span><i><span style=\"font-weight: 400;\">n<\/span><\/i><span style=\"font-weight: 400;\">, the same remainder appears in spacings of <\/span><i><span style=\"font-weight: 400;\">n<\/span><\/i><span style=\"font-weight: 400;\">? For example, 5, 9, 13, 17, 21, 25 all have remainder 1 when dividing by 4, and they\u2019re all 4 apart. This is another rule worth tucking away:<\/span><\/p>\n<p style=\"padding-left: 30px;\"><span style=\"font-weight: 400;\">If an integer <\/span><i><span style=\"font-weight: 400;\">x<\/span><\/i><span style=\"font-weight: 400;\"> has a remainder <\/span><i><span style=\"font-weight: 400;\">r<\/span><\/i><span style=\"font-weight: 400;\"> when divided by <\/span><i><span style=\"font-weight: 400;\">n<\/span><\/i><span style=\"font-weight: 400;\">, then <\/span><i><span style=\"font-weight: 400;\">x<\/span><\/i><span style=\"font-weight: 400;\"> +\/- any multiple of <\/span><i><span style=\"font-weight: 400;\">n<\/span><\/i><span style=\"font-weight: 400;\"> will also have remainder <\/span><i><span style=\"font-weight: 400;\">r<\/span><\/i><span style=\"font-weight: 400;\">.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">It\u2019s good to be familiar with this general concept of the remainder cycle, but there are two particular varieties the GMAT really gravitates towards. One of them you\u2019re likely very aware of, though you probably have never thought about it in this way. Here\u2019s a hint: it has to do with the remainder cycle when dividing by 2.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">You can see that the only possible remainders when dividing by 2 are 0 and 1. We have names for these kinds of numbers: we call them \u2018even\u2019 or \u2018odd.\u2019 Odd numbers are integers that have remainder 1 when divided by 2, and even numbers have remainder 0 when divided by 2 (that is, they are multiples of 2). You know\u2014or you will soon, as you study for the GMAT\u2014that even and odd numbers are subject to certain rules and patterns when you add, subtract, multiply, or divide them (for example: even*odd = even, odd*odd = odd, odd +\/- even = odd). The GMAT has many questions dealing with \u2018even and oddness,\u2019 though very rarely in the terms of remainders\u2014at least explicitly. But it turns out the remainder cycle is behind all the patterns in odds and evens. <\/span><\/p>\n<p><span style=\"font-weight: 400;\">However, the GMAT also has a secret affinity for the remainder cycle when dividing by 3. Now there are no labels like \u2018even\u2019 and \u2018odd\u2019 for these numbers, so I like to use \u2018R0,\u2019 \u2018R1,\u2019 and \u2018R2\u2019 for numbers that have remainder 0, 1, and 2 when dividing by 3 (R0s will be multiples of 3). Just like even and odd are the only \u2018types\u2019 of numbers when dividing by 2, these are the only \u2018types\u2019 of numbers when dividing by 3, and there are rules and patterns for these numbers as well. For instance, R1 + R2 = R0; R1 +\/- R0 = R1; R2 * R2 = R1, R2 + R2 = R1, R0 * (any integer) = R0. There are several more, but you don\u2019t need to find and memorize them; just know that they exist, and any time a problem asks about divisibility by 3, realize you\u2019ll probably need to consider these three types of numbers. <\/span><\/p>\n<p><span style=\"font-weight: 400;\">Here\u2019s a practice problem. <\/span><\/p>\n<p><i><span style=\"font-weight: 400;\">If n is an integer, which of the following must be divisible by 3?<\/span><\/i><\/p>\n<p><i><span style=\"font-weight: 400;\">A) n^3 &#8211; 4n<br \/>\n<\/span><\/i><i><span style=\"font-weight: 400;\">B) n^3 + 4n<br \/>\n<\/span><\/i><i><span style=\"font-weight: 400;\">C) n^2 +1<br \/>\n<\/span><\/i><i>D) n^2 -1<br \/>\n<\/i><i><span style=\"font-weight: 400;\">E) n^2 -4 <\/span><\/i><\/p>\n<p><span style=\"font-weight: 400;\">Since there are variables in the answer choices, you might realize you can plug in numbers. But how do you decide which? Well, since we\u2019re asked about divisibility by 3, we should probably plug in an R0, R1, and R2 (just like how on problems about even\/odd, it\u2019s best to plug in an even and an odd). <\/span><\/p>\n<p><span style=\"font-weight: 400;\">If you plug in one of each type, the only answer choice that is always divisible by 3 is A. <\/span><\/p>\n<p><span style=\"font-weight: 400;\">And here\u2019s one final bonus fact about remainders: <\/span><\/p>\n<p style=\"padding-left: 30px;\"><span style=\"font-weight: 400;\">If integer <\/span><i><span style=\"font-weight: 400;\">x<\/span><\/i><span style=\"font-weight: 400;\"> has remainder <\/span><i><span style=\"font-weight: 400;\">r<\/span><\/i><span style=\"font-weight: 400;\"> when divided by <\/span><i><span style=\"font-weight: 400;\">n<\/span><\/i><span style=\"font-weight: 400;\">, then the remainder when <\/span><i><span style=\"font-weight: 400;\">px<\/span><\/i><span style=\"font-weight: 400;\"> is divided by <\/span><i><span style=\"font-weight: 400;\">n<\/span><\/i><span style=\"font-weight: 400;\"> is equal to the remainder when <\/span><i><span style=\"font-weight: 400;\">pr<\/span><\/i><span style=\"font-weight: 400;\"> is divided by <\/span><i><span style=\"font-weight: 400;\">n<\/span><\/i><span style=\"font-weight: 400;\">.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">For example: 13 has remainder 1 when divided by 6. What is the remainder when 13 * 31 is divided by 6? Well, the remainder of 13 * 31 will be the same as the remainder of 1 * 31 when dividing by 6. And the remainder of 31 when divided by 6 is 1. <\/span><\/p>\n<p><span style=\"font-weight: 400;\">So welcome back to middle school! Now that remainders have returned to your life, social ostracism, acne, and raging hormones are sure to follow. ?<\/span><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/giphy.com\/embed\/mEqMknMZWh1Fm\" width=\"480\" height=\"271\" frameBorder=\"0\" class=\"giphy-embed\" allowFullScreen><\/iframe><\/p>\n<p><a href=\"https:\/\/giphy.com\/gifs\/chrissy-nancy-odell-mEqMknMZWh1Fm\"><\/a><\/p>\n<hr \/>\n<p><b><i>Want some more GMAT tips from Reed? Attend the first session of one of his\u00a0<\/i><\/b><a id=\"bloglink\" href=\"https:\/\/www.manhattanprep.com\/gmat\/classes\/#instructor\/366\" target=\"_blank\" rel=\"noopener\"><b><i>upcoming GMAT courses\u00a0<\/i><\/b><\/a><b><i>absolutely free, no strings attached. Seriously.<\/i><\/b><\/p>\n<hr \/>\n<p><em><strong><a id=\"bloglink\" href=\"https:\/\/www.manhattanprep.com\/instructors\/reed-arnold\/\" target=\"_blank\" rel=\"noopener\">Reed Arnold<\/a><a href=\"https:\/\/www.manhattanprep.com\/instructors\/reed-arnold\/?utm_source=manhattanprep.com%2Fgmat%2Fblog&#038;utm_campaign=GMAT%20Blog&#038;utm_medium=blog&#038;utm_content=Reed%20Arnold%20Bio%20Link\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-12946 size-thumbnail\" src=\"https:\/\/d27gmszdzgfpo3.cloudfront.net\/gmat\/wp-content\/uploads\/sites\/18\/2017\/01\/reed-arnold-150x150.png\" width=\"150\" height=\"150\" data-pin-nopin=\"true\" \/><\/a>\u00a0is a Manhattan Prep instructor based in New York, NY.<\/strong>\u00a0He has a B.A. in economics, philosophy, and mathematics and an M.S. in commerce, both from the University of Virginia. He enjoys writing, acting, Chipotle burritos, and teaching the GMAT.\u00a0<a id=\"bloglink\" href=\"https:\/\/www.manhattanprep.com\/gmat\/classes\/#instructor\/366\" target=\"_blank\" rel=\"noopener\">Check out Reed\u2019s upcoming GMAT courses here.<\/a><\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Guess what? You can attend the first session of any of our online or in-person GMAT courses absolutely free\u2014we\u2019re not kidding! Check out our upcoming courses here. One common complaint I hear from my students is that they \u2018haven\u2019t done math like this since high school.\u2019 And they\u2019re pretty much right: the concepts in the [&hellip;]<\/p>\n","protected":false},"author":158,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11,873,18,880,929,874,52871,930,2,8],"tags":[639,53156],"yst_prominent_words":[],"class_list":["post-14702","post","type-post","status-publish","format-standard","hentry","category-manhattan-gmat-blog-algebra","category-for-current-studiers","category-manhattan-gmat-blog-fdp","category-gmat-101","category-gmat-prep","category-gmat-resources","category-gmat-strategies","category-gmat-study-guide","category-how-to-study","category-quant-on-gmat","tag-remainders","tag-the-remainder-cycle"],"_links":{"self":[{"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/posts\/14702","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/users\/158"}],"replies":[{"embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/comments?post=14702"}],"version-history":[{"count":6,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/posts\/14702\/revisions"}],"predecessor-version":[{"id":14771,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/posts\/14702\/revisions\/14771"}],"wp:attachment":[{"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/media?parent=14702"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/categories?post=14702"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/tags?post=14702"},{"taxonomy":"yst_prominent_words","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/yst_prominent_words?post=14702"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}