{"id":11127,"date":"2016-03-09T16:17:31","date_gmt":"2016-03-09T16:17:31","guid":{"rendered":"http:\/\/www.manhattanprep.com\/gmat\/?p=11127"},"modified":"2019-11-08T19:45:20","modified_gmt":"2019-11-08T19:45:20","slug":"decoding-divisibility-and-primes-on-the-gmat-part-3","status":"publish","type":"post","link":"https:\/\/www.manhattanprep.com\/gmat\/blog\/decoding-divisibility-and-primes-on-the-gmat-part-3\/","title":{"rendered":"Decoding Divisibility and Primes on the GMAT \u2013 Part 3"},"content":{"rendered":"

\"DecodingDid you know that you can attend the first session of any of our online or in-person GMAT courses absolutely free? We\u2019re not kidding! Check out our upcoming courses here<\/a>.<\/em><\/strong><\/p>\n

\n

Welcome to the 3rd<\/sup> and final installment of our mini-series on GMAT Number Properties. We\u2019ve been doing a deep dive into Divisibility and Prime issues (see Part 1<\/a> and Part 2<\/a>), exploring how the GMAT can disguise these topics and get us to fall into traps.<\/p>\n

Here\u2019s your third GMATPrep\u00ae problem from the free exams\u2014and our hardest one yet. Good luck!<\/p>\n

\u201c*For every positive even integer n<\/i>, the function h(n)<\/i> is defined to be the product of all the even integers from 2 to n<\/i>, inclusive. If p<\/i> is the smallest prime factor of h<\/i>(100) + 1, then p<\/i> is<\/p>\n

\u201c(A) between 2 and 10<\/p>\n

\u201c(B) between 10 and 20<\/p>\n

\u201c(C) between 20 and 30<\/p>\n

\u201c(D) between 30 and 40<\/p>\n

\u201c(E) greater than 40\u201d<\/p><\/blockquote>\n

Yikes. Where to start?<\/strong><\/p>\n

\"Glance<\/a><\/p>\n

Glance<\/i>: PS. A lot of words. The answers are in a weird form. But maybe that means I can estimate?<\/p>\n

Read<\/i>: Ugh: functions. I need to jot this down while I\u2019m reading in order to figure out what it means.<\/p>\n

Jot<\/i>:<\/p>\n

\"Decoding\"\"<\/p>\n

Okay. So the function equals the evens multiplied together\u2026<\/p>\n

h<\/i>(100) = (2)(4)(6)(\u2026)(100)\u2026 And then I have to add 1.<\/p>\n

\"Reflect<\/a><\/p>\n

What does that mean? I\u2019m not really sure. I have to think about this using a real number. 2 is the smallest prime number period; could it be a prime factor of that h<\/i>(100) + 1 thing?<\/p>\n

Let\u2019s see. 2 is a factor of h<\/i>(100), because h<\/i>(100) = (2)(4)(6)(\u2026)(100).<\/p>\n

And then if I add 1\u2026oh, 2 can\u2019t be a factor if I add 1! That would make h<\/i>(100) odd, and anything odd doesn\u2019t have 2 as a factor! Okay.<\/p>\n

I can\u2019t keep testing every prime, of course, but I haven\u2019t found a pattern yet. There must be a pattern, since nobody could really calculate this number without a computer. So I think I\u2019ve found my path; I\u2019m going to try 3 next.<\/p>\n

\"GRJ<\/a><\/p>\n

Hmm. Oh, check it out: h<\/i>(100) = (2)(4)(6)(\u2026)(100). 6 has 3 as a factor, so 3 is a factor of h<\/em>(100), too. And then I add 1\u2026<\/p>\n

I think I see! So if that big number has 3 as a factor and then I add just 1, the new number can\u2019t also be a factor of 3. I would have to add another 3 to get the new number to also be a factor of 3, since all multiples of 3 are at least 3 apart.<\/p>\n

In other words, if h<\/i>(100) has a factor of 3, then h<\/i>(100) + 3 has a factor of 3, but h<\/i>(100) + 1 does not. I have to add at least the factor itself in order for the new number to have that same factor.<\/p>\n

Okay, I\u2019m excited. This might be the pattern\u2026<\/p>\n

h<\/i>(100) = (2)(4)(6)(\u2026)(100)<\/p>\n

This multiplication is in the form (2)(1), (2)(2), (2)(3), (2)(4), and so on up to (2)(50). In other words, every integer between 1 and 50 is part of the multiplication at least once, so every integer between 1 and 50 is a factor of h<\/i>(100), including all the prime numbers between 1 and 50.<\/p>\n

If any given prime is a factor of h<\/i>(100), then that prime cannot<\/i> be a factor of h<\/i>(100) + 1, because adding 1 isn\u2019t enough. For example, if h<\/i>(100) has a factor of 5, then h<\/i>(100) + 5 has a factor of 5, but h<\/i>(100) + 1 does not. And so on.<\/p>\n

All the primes up to 50 are covered, so p<\/i> has to be something greater than 50.<\/p>\n

The correct answer is (E).<\/strong><\/p>\n

If you want to know, 53 is the first prime number after 50, so the smallest possible value for p<\/i> is 53. But this doesn\u2019t mean that p<\/em> is 53! In fact, it\u2019d be pretty impossible to figure out what p<\/i> is without a calculator or computer program and in only 2 minutes. Sure, a human calculator can probably do it, but business schools aren\u2019t actually looking for human calculators (even though it can feel like that sometimes!). That\u2019s why the question doesn\u2019t ask what p<\/i> is.<\/p>\n

Key Takeaways for Divisibility and Primes on the GMAT:<\/strong><\/p>\n

(1) When the math looks ridiculous\u2026that\u2019s because it is. \":)\" Look for the pattern. If you can find it, as we did above, then you may be able to solve. If you can\u2019t, shrug your shoulders, be glad that you earned such a hard question in the first place, pick your favorite letter, and move on.<\/p>\n

(2) As you study Number Properties, start to train yourself to read the clues: what are they hiding from you in the language or setup? In all three problems, we had to manipulate or think about the math given in a way that we never learned in school. For divisibility and primes, this often revolves around what factors multiplied by what other factors do\u2014or do not, in the case of Data Sufficiency\u2014get us to the answer.<\/p>\n

* GMATPrep\u00ae questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.<\/p>\n

\n

Can\u2019t get enough of Stacey\u2019s GMAT mastery? Attend the first session of one of her upcoming GMAT courses<\/a> absolutely free, no strings attached. Seriously.\u00a0<\/strong><\/em><\/p>\n

\n

\"stacey-koprince\"<\/a>Stacey Koprince<\/a> is a Manhattan Prep instructor based in Montreal, Canada and Los Angeles, California.<\/strong> Stacey has been teaching the GMAT, GRE, and LSAT \u00a0for more than 15 years and is one of the most well-known instructors in the industry. Stacey loves to teach and is absolutely fascinated by standardized tests. Check out Stacey\u2019s upcoming GMAT courses here<\/a>.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"

Did you know that you can attend the first session of any of our online or in-person GMAT courses absolutely free? We\u2019re not kidding! Check out our upcoming courses here. Welcome to the 3rd and final installment of our mini-series on GMAT Number Properties. We\u2019ve been doing a deep dive into Divisibility and Prime issues […]<\/p>\n","protected":false},"author":6,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[873,929,24,8],"tags":[52784,52760],"yst_prominent_words":[56472,56060,56471,53868,53779,56474,53781,53775,56466,56468,56473,56470,54189,55065,53867],"class_list":["post-11127","post","type-post","status-publish","format-standard","hentry","category-for-current-studiers","category-gmat-prep","category-problem-solving","category-quant-on-gmat","tag-gmat-divisibility-and-primes","tag-gmat-number-properties"],"_links":{"self":[{"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/posts\/11127","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\/6"}],"replies":[{"embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/comments?post=11127"}],"version-history":[{"count":3,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/posts\/11127\/revisions"}],"predecessor-version":[{"id":18278,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/posts\/11127\/revisions\/18278"}],"wp:attachment":[{"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/media?parent=11127"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/categories?post=11127"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/tags?post=11127"},{"taxonomy":"yst_prominent_words","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gmat\/wp-json\/wp\/v2\/yst_prominent_words?post=11127"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}