{"id":3775,"date":"2013-07-17T09:11:15","date_gmt":"2013-07-17T16:11:15","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=3775"},"modified":"2020-01-15T10:49:01","modified_gmt":"2020-01-15T18:49:01","slug":"gmat-quant-difficult-units-digits-questions","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/","title":{"rendered":"GMAT Quant: Difficult Units Digits Questions"},"content":{"rendered":"

\"Girl<\/p>\n

First, a couple 800+ practice questions (yes, you read that right – 800+) on which to whet your chops.<\/p>\n

1) The units digit of \"(137^13)^47\" \u00a0is:<\/p>\n

(A) 1<\/p>\n

(B) 3<\/p>\n

(C) 5<\/p>\n

(D) 7<\/p>\n

(E) 9<\/p>\n

2) The units digit of \"35^87+93^46\" is:<\/p>\n

(A) 2<\/p>\n

(B) 4<\/p>\n

(C) 6<\/p>\n

(D) 8<\/p>\n

(E) 0<\/p>\n

3) The units digit of \"(44^91)*(73^37)\" is:<\/p>\n

(A) 2<\/p>\n

(B) 4<\/p>\n

(C) 6<\/p>\n

(D) 8<\/p>\n

(E) 0<\/p>\n

Admittedly, these problems are probably a notch harder than anything you are likely to see on the GMAT.\u00a0 If you understand these, you definitely will understand anything of this variety that the GMAT throws at you!<\/p>\n

 <\/p>\n

Not what it seems<\/h2>\n

All of those problems above involve numbers with hundreds of decimal places.\u00a0 No one can calculate those answers without a calculator: in fact, no calculator would be sufficient to do the calculation, because no calculator can accommodate that many digits.\u00a0 If one needed the exact answer, one could always use that most extraordinary web computing tool, Wolfram Alpha<\/a>.\u00a0 Of course, one will not have access to the web or a calculator<\/a> or anything other than one’s owns wits when confronting a question such as this on the GMAT.\u00a0 How do we proceed?<\/p>\n

It turns out, what appears as a ridiculously hard calculation is actually quite easier.\u00a0 No part of the calculation we are going to do will involve anything beyond single-digit arithmetic!<\/p>\n

Units digit arithmetic<\/h2>\n

The units digits of large numbers are special: they form a kind of elite and exclusive club.\u00a0 The big idea: only units digits affect units digits<\/b>.\u00a0 What do I mean by that?\u00a0 Well, first of all, suppose you add or subtract two large numbers —- the units digit of the sum or the difference will depend only on the units digits of the two input numbers<\/b>.\u00a0 For example, 3 + 5 = 8 —- this means that any number ending in 3<\/i> plus any number ending in 5<\/i> will be a number ending in 8.\u00a0 If you remember your “column addition” processes from grade school, this one might make intuitive sense.<\/p>\n

The one that can be a little harder for folks to swallow is multiplication.\u00a0 If you multiply two large numbers, the unit digit of the product will have the same units digit as the product of the units digit of the two factors<\/b>.\u00a0 That’s a mouthful!\u00a0 In other words, let’s take 3*7 = 21, so a units digit of a 3, times a unit digit of a 7, equals a units digit of a 1.\u00a0 That means, we could take any large number ending in 3, times any large number ending in 7, and the product absolutely will have to have a units digit of 1.\u00a0 If this is new idea to you, I strongly recommend: sit down with a calculator and multiply ridiculously large numbers together, with all combinations of units digits, until you are 100% satisfied that this pattern works.<\/p>\n

 <\/p>\n

Units digits and powers<\/h2>\n

This part will be a recap of an earlier post on powers of units digits<\/a>.\u00a0 When we raise to a power, of course, that’s iterated multiplication, so we just follow the multiplication rule above.\u00a0 As it turns out, a simple pattern will always emerge.<\/p>\n

Suppose we were considering powers of 253 — first of all, only the units digit, 3, matters, for the units digit of the powers.\u00a0 Any number ending in three will have the same sequence of units digits for the powers.<\/p>\n

\"3^1\" = 3<\/p>\n

\"3^2\" = 9<\/p>\n

9*3 = 27, so \"3^3\" has a units digit of 7<\/p>\n

7*3 = 21, so \"3^4\" has a units digit of 1<\/p>\n

1*3 = 3, so \"3^5\" has a units digit of 3<\/p>\n

3*3 = 9, so \"3^6\" has a units digit of 9<\/p>\n

9*3 = 27, so \"3^7\" has a units digit of 7<\/p>\n

7*3 = 21, so \"3^8\" has a units digit of 1<\/p>\n

Notice a pattern has emerged — 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, \u2026\u00a0 It repeats like mathematical wallpaper.\u00a0\u00a0 The pattern has a period<\/b> of 4 — that is to say, it take four steps to repeat.\u00a0 This means, 3 to the power of any multiple of 4 has a units digit of 1: \"3^8\", \"3^56\", and \"3^444\" all have a units digit of one.\u00a0 If I want to find power that’s not a multiple of 4, that’ easy: I just go to the nearest multiple of 4 and follow the wallpaper from there.\u00a0\u00a0 For example, if I wanted \"3^446\" —–<\/p>\n

\"3^444\" has a units digit of 1<\/p>\n

\"3^445\" has a units digit of 3<\/p>\n

\"3^446\" has a units digit of 9<\/p>\n

As it happens, \"3^446\" is a number that has 213 digits, but the units digit must be a 9.<\/p>\n

The really expansive idea: everything I have just said about powers of 3 also applies to any larger number that happens to have a units digit of 3.\u00a0 Thus,<\/p>\n

\"253^446\" has a units digit of 9<\/p>\n

That number has over a thousand digits (you don’t need to know how to figure that out!) but we know for sure that the units digit of this gargantuan number is 9.<\/p>\n

 <\/p>\n

Summary<\/h2>\n

If reading this article gave you any insights, you may want to give the questions above a second try.\u00a0 Here’s another problem, slightly easier and more GMAT-like, of this genre:<\/p>\n

4) http:\/\/gmat.magoosh.com\/questions\/648<\/a><\/p>\n

If there is anything you would like to say on this topic, or if you have any questions, please let me know in the comments section.<\/p>\n

Finally, on a totally gratuitous note, here, in it’s the full thousand-plus-digit glory, is \"253^446\", courtesy of Wolfram Alpha<\/a>:<\/p>\n

\"253^446\" = 6190880832531899190821690500833264378796558621138440416684569816896956548548008694
\n5501637120599244459832600741641042834529231770919235178215551804306285786976623543
\n72380800482154741117883167160446214356675850891464689434900627878445424968534812213
\n04515041521698298553935861735825956938171070649017829532068323699186643091574519875
\n641384511684642401730622089635116285314952964987090639595147866795944410916642166093
\n585848058327971863158257619930226042698661632905146850162960633520155118628867911625
\n239725418415604877699453370534194837316774432534349898272185517986005836675979188507
\n704257742239368667474408667895362250511057160490511029003928521905584001998500412272
\n300652930331121107733643816582958394189572596322595033481338694429893546070448926193
\n272806103607662243076062238492673013489615386273692928543218155895489937520257687664
\n947027647847750945509362588170852889875925160078794611182855714905968753089053225774
\n863189760920183769031795458368827168630624310066175033265292467587132663811805301906
\n7641362643313498166787213628751583911774745199740840719668395714479929<\/p>\n

Notice, of course, that the units digit is 9.<\/p>\n

 <\/p>\n

Practice problem solutions<\/h2>\n

1) First of all, all we need is the last digit of the base, not 137, but just 7.\u00a0 Here’s the power sequence of the units of 7<\/p>\n

\"7^1\" has a units digit of 7<\/p>\n

\"7^2\" has a units digit of 9\u00a0 (e.g. 7*7 = 49)<\/p>\n

\"7^3\" has a units digit of 3\u00a0 (e.g. 7*9 = 63)<\/p>\n

\"7^4\" has a units digit of 1\u00a0 (e.g. 7*3 = 21)<\/p>\n

\"7^5\" has a units digit of 7<\/p>\n

\"7^6\" has a units digit of 9<\/p>\n

\"7^7\" has a units digit of 3<\/p>\n

\"7^8\" has a units digit of 1<\/p>\n

etc.<\/p>\n

The period is 4, so 7 to the power of any multiple of 4 has a units digit of 1<\/p>\n

\"7^12\" has a units digit of 1<\/p>\n

\"7^13\" has a units digit of 7<\/p>\n

So the inner parenthesis is a number with a units digit of 7.<\/p>\n

Now, for the outer exponent, we are following the same pattern — starting with a units digit of 7.\u00a0 The period is still 4.<\/p>\n

\"7^44\" has a units digit of 1<\/p>\n

\"7^45\" has a units digit of 7<\/p>\n

\"7^46\" has a units digit of 9<\/p>\n

\"7^47\" has a units digit of 3<\/p>\n

So the unit digit of the final output is 3.\u00a0 Answer = B<\/b><\/p>\n

BTW, this number is the great-granddaddy, the biggest number of all the big numbers mentioned in this post.\u00a0 The number \"(137^13)^47\" clocks in with over 1300 digits!<\/p>\n

 <\/p>\n

2) We have to figure out each piece separately, and then add them.\u00a0 The first piece is remarkably easy — any power of anything ending in 5 always has a units digit of 5.\u00a0 So the first term has a units digit of 5.\u00a0 Done.<\/p>\n

The second term takes a little more work.\u00a0\u00a0 We can ignore the tens digit, and just treat this base as 3. \u00a0Here is the units digit patter for the powers of 3.<\/p>\n

\"3^1\" has a units digit of 3<\/p>\n

\"3^2\" has a units digit of 9<\/p>\n

\"3^3\" has a units digit of 7\u00a0 (e.g. 3*9 = 27)<\/p>\n

\"3^4\" has a units digit of 1\u00a0 (e.g. 3*7 = 21)<\/p>\n

\"3^5\" has a units digit of 3<\/p>\n

\"3^6\" has a units digit of 9<\/p>\n

\"3^7\" has a units digit of 7<\/p>\n

\"3^8\" has a units digit of 1<\/p>\n

The period is 4.\u00a0 This means, 3 to the power of any multiple of 4 will have a units digit of 1.<\/p>\n

\"3^44\" has a units digit of 1<\/p>\n

\"3^45\" has a units digit of 3<\/p>\n

\"3^46\" has a units digit of 9<\/p>\n

Therefore, the second term has a units digit of 9.<\/p>\n

Of course 5 + 9 = 14, so something with a units digit of 5 plus something with a units digit of 9 will have a units digit of 4.\u00a0 Answer = B<\/b><\/p>\n

 <\/p>\n

3) We have to figure out each piece separately, and then multiply them.\u00a0 The powers of 4 are particularly easy.<\/p>\n

\"4^1\" has a units digit of 4<\/p>\n

\"4^2\" has a units digit of 6\u00a0 (e.g. 4*4 = 16)<\/p>\n

\"4^3\" has a units digit of 4\u00a0 (e.g. 4*6 = 24)<\/p>\n

\"4^4\" has a units digit of 6<\/p>\n

\"4^5\" has a units digit of 4<\/p>\n

\"4^6\" has a units digit of 6<\/p>\n

Four to any odd power will have a units digit of 4.\u00a0 Thus, any number with a units digit of four, raised to an odd power, will also have a units digit of 4.\u00a0 The first factor, \"44^91\", has a units digit of 4.<\/p>\n

Now, the base in the second factor ends in a 3 (we can ignore the tens digit).\u00a0 Here is the pattern for powers of three.<\/p>\n

\"3^1\" has a units digit of 3<\/p>\n

\"3^2\" has a units digit of 9<\/p>\n

\"3^3\" has a units digit of 7\u00a0 (e.g. 3*9 = 27)<\/p>\n

\"3^4\" has a units digit of 1\u00a0 (e.g. 3*7 = 21)<\/p>\n

\"3^5\" has a units digit of 3<\/p>\n

\"3^6\" has a units digit of 9<\/p>\n

\"3^7\" has a units digit of 7<\/p>\n

\"3^8\" has a units digit of 1<\/p>\n

The period is 4. \u00a0This means, 3 to the power of any multiple of 4 will have a units digit of 1.<\/p>\n

\"3^36\" has a units digit of 1<\/p>\n

\"3^37\" has a units digit of 3<\/p>\n

Thus, any number with a units digit of 7, when raised to the power of 37, will have a units digit of 3.\u00a0 The second factor, \"73^37\", has a units digit of 3.<\/p>\n

Of course, 4*3 = 12, so any number with a units digit of 4 times any number with a units digit of 3 will yield a product with a units digit of 2.<\/p>\n

Answer = A<\/b><\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

First, a couple 800+ practice questions (yes, you read that right – 800+) on which to whet your chops. 1) The units digit of \u00a0is: (A) 1 (B) 3 (C) 5 (D) 7 (E) 9 2) The units digit of is: (A) 2 (B) 4 (C) 6 (D) 8 (E) 0 3) The units digit […]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[160],"tags":[],"ppma_author":[13209],"class_list":["post-3775","post","type-post","status-publish","format-standard","hentry","category-arithmetic"],"acf":[],"yoast_head":"\nGMAT Quant: Difficult Units Digits Questions - Magoosh Blog \u2014 GMAT\u00ae Exam<\/title>\n<meta name=\"description\" content=\"What do you need to know about unit digit problems for GMAT Quant? Get the expert answer here, along with practice problems and explanations.\" \/>\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-difficult-units-digits-questions\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"GMAT Quant: Difficult Units Digits Questions\" \/>\n<meta property=\"og:description\" content=\"What do you need to know about unit digit problems for GMAT Quant? Get the expert answer here, along with practice problems and explanations.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/\" \/>\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=\"2013-07-17T16:11:15+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2020-01-15T18:49:01+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/magoosh.com\/gmat\/files\/2013\/07\/image-gmat-header-addingSubtracting.jpg\" \/>\n<meta name=\"author\" content=\"Mike M\u1d9cGarry\" \/>\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=\"Mike M\u1d9cGarry\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"8 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-difficult-units-digits-questions\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/\"},\"author\":{\"name\":\"Mike M\u1d9cGarry\",\"@id\":\"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/320346c205075513344435baf9b0521b\"},\"headline\":\"GMAT Quant: Difficult Units Digits Questions\",\"datePublished\":\"2013-07-17T16:11:15+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/\"},\"wordCount\":1604,\"commentCount\":38,\"publisher\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/#organization\"},\"articleSection\":[\"GMAT Arithmetic\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/\",\"url\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/\",\"name\":\"GMAT Quant: Difficult Units Digits Questions - Magoosh Blog \u2014 GMAT\u00ae Exam\",\"isPartOf\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/#website\"},\"datePublished\":\"2013-07-17T16:11:15+00:00\",\"description\":\"What do you need to know about unit digit problems for GMAT Quant? Get the expert answer here, along with practice problems and explanations.\",\"breadcrumb\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/magoosh.com\/gmat\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"GMAT Quant: Difficult Units Digits Questions\"}]},{\"@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\/320346c205075513344435baf9b0521b\",\"name\":\"Mike M\u1d9cGarry\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/image\/15a1e36ef1c2c3940179212433de141a\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g\",\"caption\":\"Mike M\u1d9cGarry\"},\"description\":\"Mike holds an A.B. in Physics (graduating magna cum laude) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations.\",\"sameAs\":[\"https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured\"],\"award\":[\"Magna cum laude from Harvard\"],\"knowsAbout\":[\"GMAT\"],\"knowsLanguage\":[\"English\"],\"jobTitle\":\"Content Creator\",\"worksFor\":\"Magoosh\",\"url\":\"https:\/\/magoosh.com\/gmat\/author\/mikemcgarry\/\"}]}<\/script>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"GMAT Quant: Difficult Units Digits Questions - Magoosh Blog \u2014 GMAT\u00ae Exam","description":"What do you need to know about unit digit problems for GMAT Quant? Get the expert answer here, along with practice problems and explanations.","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-difficult-units-digits-questions\/","og_locale":"en_US","og_type":"article","og_title":"GMAT Quant: Difficult Units Digits Questions","og_description":"What do you need to know about unit digit problems for GMAT Quant? Get the expert answer here, along with practice problems and explanations.","og_url":"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/","og_site_name":"Magoosh Blog \u2014 GMAT\u00ae Exam","article_publisher":"https:\/\/www.facebook.com\/MagooshGMAT\/","article_published_time":"2013-07-17T16:11:15+00:00","article_modified_time":"2020-01-15T18:49:01+00:00","og_image":[{"url":"https:\/\/magoosh.com\/gmat\/files\/2013\/07\/image-gmat-header-addingSubtracting.jpg"}],"author":"Mike M\u1d9cGarry","twitter_card":"summary_large_image","twitter_creator":"@MagooshGMAT","twitter_site":"@MagooshGMAT","twitter_misc":{"Written by":"Mike M\u1d9cGarry","Est. reading time":"8 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/#article","isPartOf":{"@id":"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/"},"author":{"name":"Mike M\u1d9cGarry","@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/320346c205075513344435baf9b0521b"},"headline":"GMAT Quant: Difficult Units Digits Questions","datePublished":"2013-07-17T16:11:15+00:00","mainEntityOfPage":{"@id":"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/"},"wordCount":1604,"commentCount":38,"publisher":{"@id":"https:\/\/magoosh.com\/gmat\/#organization"},"articleSection":["GMAT Arithmetic"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/","url":"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/","name":"GMAT Quant: Difficult Units Digits Questions - Magoosh Blog \u2014 GMAT\u00ae Exam","isPartOf":{"@id":"https:\/\/magoosh.com\/gmat\/#website"},"datePublished":"2013-07-17T16:11:15+00:00","description":"What do you need to know about unit digit problems for GMAT Quant? Get the expert answer here, along with practice problems and explanations.","breadcrumb":{"@id":"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/magoosh.com\/gmat\/gmat-quant-difficult-units-digits-questions\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/magoosh.com\/gmat\/"},{"@type":"ListItem","position":2,"name":"GMAT Quant: Difficult Units Digits Questions"}]},{"@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\/320346c205075513344435baf9b0521b","name":"Mike M\u1d9cGarry","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/image\/15a1e36ef1c2c3940179212433de141a","url":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","caption":"Mike M\u1d9cGarry"},"description":"Mike holds an A.B. in Physics (graduating magna cum laude) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations.","sameAs":["https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured"],"award":["Magna cum laude from Harvard"],"knowsAbout":["GMAT"],"knowsLanguage":["English"],"jobTitle":"Content Creator","worksFor":"Magoosh","url":"https:\/\/magoosh.com\/gmat\/author\/mikemcgarry\/"}]}},"authors":[{"term_id":13209,"user_id":26,"is_guest":0,"slug":"mikemcgarry","display_name":"Mike M\u1d9cGarry","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","user_url":"","last_name":"M\u1d9cGarry","first_name":"Mike","description":"Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as \"member of the month\" for over two years at <a href=\"https:\/\/gmatclub.com\/blog\/2012\/09\/mike-mcgarrys-gmat-experience\/\" rel=\"noopener noreferrer\">GMAT Club<\/a>. Mike holds an A.B. in Physics (graduating <em>magna cum laude<\/em>) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's <a href=\"https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured\" rel=\"noopener noreferrer\">Youtube <\/a>video explanations and resources like <a href=\"https:\/\/magoosh.com\/gmat\/whats-a-good-gmat-score\/\" rel=\"noopener noreferrer\">What is a Good GMAT Score?<\/a> and the <a href=\"https:\/\/magoosh.com\/gmat\/gmat-diagnostic-test\/\" rel=\"noopener noreferrer\">GMAT Diagnostic Test<\/a>."}],"_links":{"self":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/3775","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\/26"}],"replies":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/comments?post=3775"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/3775\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/media?parent=3775"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/categories?post=3775"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/tags?post=3775"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/ppma_author?post=3775"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}