Actually, we can find the last two digits quite easily in most such cases by using the concepts of remainders.

There are two concepts you need to understand before we go on to see how to solve such questions:

I. **When you divide a number by 100, the remainder is formed by the last two digits of the number.** Say, you divide 138 by 100, the remainder will be 38 (last two digits). Take another example – divide 1275 by 100, the remainder will be 75 and so on.

II. **When you divide (px + a)(qx + b)*…*(tx + e) by x, the remainder will be the remainder obtained by dividing a*b*…*e by x**. This should remind you of the binomial theorem we discussed many weeks ago. When we multiply all these terms together (px + a), (qx + b) etc, each term obtained will have at least one x except the last term which is obtained by multiplying the remainders together. To get a better idea, let’s take some numbers:

Let’s say we need to find the remainder when we divide 12*23*52*81 by 10.

K = 12*23*52*81 = (10 + 2)*(20 + 3)*(50 +2)*(80 + 1)

When you multiply these four terms together, you will get many terms such as 10*20*50*80, 10*20*50*1, 10*20*2*80 etc. All these will have a multiple of 10 except the last one. The last one will be 2*3*2*1 = 12. That doesn’t have a multiple of 10. Now divide 12 by 10 to get the remainder 2. So when you divide K by 10, the remainder will be 2.

Now, let’s look at a question:

Question 1: What are the last two digits of 63*35*37*82*71*41?

(A) 10

(B) 30

(C) 40

(D) 70

(E) 80

Solution: Using concept 1, we know that to find the last two digits, we need to find the remainder we get when we divide the product by 100.

Remainder of (63*35*37*82*71*41)/ 100

Note that we can simplify this expression by canceling out the 5 and 2 in the numerator and denominator. But before we do that, here is an important note:

*Note: We cannot just cancel off the common terms in the numerator and denominator to get the remainder. But, if we want to cancel off to simplify the question, we can do it, provided we remember to multiply it back again.
So say, we want to find the remainder when 14 is divided by 10 i.e. 14/10 (remainder 4). But we cancel off the common 2 to get 7/5. The remainder here will be 2 which is not the same as the remainder obtained by dividing 14 by 10. But if we multiply 2 back by 2 (the number we canceled off), the remainder will become 2*2 = 4 which is correct.
Take another example to reinforce this – what is the remainder when 85 is divided by 20? It is 5.
We might rephrase it as – what is the remainder when 17 is divided by 4 (cancel off 5 from the numerator and the denominator). The remainder in this case is 1. We multiply the 5 back to 1 to get the remainder as 5 which is correct.*

So keeping this very important point in mind, let’s go ahead and cancel the common 5 and 2.

We need the

Remainder of (63*7*37*41*71*41*5*2)/10*5*2

Remainder of (63*7*37*41*71*41)/10

Now using concept 2, let’s write the numbers in form of multiples of 10

Remainder of (60+3)*7*(30+7)*(40+1)*(70+1)*(40+1)/10

Remainder of 3*7*7*1*1*1/10

Remainder of 147/10 = 7

Now remember, we had canceled off 10 so to get the actual remainder so we need to multiply by 10: 7*10 = 70.

When 63*35*37*82*71*41 is divided by 100, the remainder is 70. So the last two digits of 63*35*37*82*71*41 must be 70.

Answer (D)

Next week, we will see some more complicated questions using these and other fundamentals.

*Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the **GMAT** for Veritas Prep and regularly participates in content development projects such as this blog!*

Routine works to improve your test score on a few levels. It gets your body physically accustomed to test day conditions as well as works to quell the nerves (your biggest enemy on test day can be anxiety). Routine is powerful! Here are a few ways in which you can establish a routine that will help you score better on test day.

**Sleep Schedule:** You want to ensure that you know exactly how your body and mind will feel on test day. Establishing a good sleep schedule in the weeks before the test will play a crucial role in this. If you’re used to sleeping in until 10am and taking your practice ACT at 11am, it will come as a shock to wake up at 6am on test day. You will know your test time weeks before the actual exam; work on getting your body accustomed to the sleep schedule that this test time requires.

**“Test” Practice:** You can work to simulate testing conditions in a number of ways. If you’re scheduled to take your ACT on Saturday at 8 am, practice waking up at 6am, driving to another location, sitting down, and taking a practice exam four Saturdays before the test. The more times you do the routine, the more comfortable you will feel on test day. If you can practice at your actual test center, that’s great! Be sure to account for timing and standardized breaks.

**Nutrition:** Food is a critical part of routine that is often forgotten. Hunger can distract your attention from the exam, so be sure to figure out what kinds of snacks you need to bring. This plays a significant role on test day, from the breakfast you eat, to the snacks you bring, to the amount of liquids you consume. Everyone is different, so it can be helpful to test out what kinds of breakfast keeps you full and satisfied throughout your exam. While it may seem trivial, making sure you stay hydrated without over-hydrating is important too; no one wants to feel uncomfortable during testing.

These three simple tips for establishing a routine will help you feel more comfortable on test day. Once you’ve established a routine, you can walk into your actual test day feeling more confident and you’ll know exactly what to expect. Happy Studying!

For more tips on acing the ACT and getting into the most competitive universities in the nation, be sure to find us on Facebook and Google+, and follow us on Twitter!

*Sarah Smith is a Pre-med, Bioethics major at Northwestern University. She’s editor in chief and co-founder of the student health magazine and enjoys being involved in various clubs around campus. Sarah is passionate about education and enjoys learning and teaching. She enjoys helping Veritas Prep students prepare for the ACT!*

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The other answer we can give you? Sarah Koenig would do pretty darned well on Data Sufficiency questions, where often it’s just as important to determine what you don’t know as it is to determine what you do. While the internet buzzed with theories certain that Adnan did it, that Jay did it, that a recently-released serial killer did it, Koenig was often ridiculed for being so noncommittal in her assessment of whether Adnan is guilty or not. But that’s an important mentality on Data Sufficiency questions, as one of the common ways that the GMAT will bait you is giving you information that seems overwhelmingly sufficient (The Nisha call! The phone was in Leakin Park!) but that leaves just enough doubt (Why did Jay’s story change so much?) that you can’t prove a definitive answer. And like the jury in the Serial case, we all have that tendency to jump to conclusions (“well if he didn’t kill her, who did?”) and filter out information that we don’t like (Christina Gutierrez’s performance…). This Serial-themed Data Sufficiency problem should exemplify (forgive the lack of subscript formatting, but a sequence problem in a Serial blog post seemed fitting):

The infinite (serial) sequence a1, a2, …, an, … is such that a1 = x, a2 = y, a3 = z,a4 = 3 and an = a(n-4) for n > 4. What is the sum of the first 98 terms of the sequence?

(1) x = 5

(2) y + z = 2

As people unpack the mystery in this problem, they start to see what’s going on. If an = a(n-4), then each term equals the term that came four prior. So the sequence really goes:

x, y, z, 3, x, y, z, 3, x, y, z, 3…

So although it looks like a pretty massive mystery, really you’re trying to figure out x, y, and z because 3 is just 3. And here’s a common way of thinking:

Statement 1 is not sufficient, but it gets you one of the terms. And Statement 2 is not sufficient but it gets you two more. So when you put them together, you know that the sum of one trip through the 4-term sequence is 5 + 2 + 3 = 10, so you should be able to extrapolate that to the whole thing, right? Just figure out how many trips through will get you to term 98 and you have it; like the Syed jury, you have the motive and the timeline and the cell phone records and Jay’s testimony, so the answer has to be C. Right?

But let’s interview Sarah Koenig here:

Sarah: The pieces all seem to fit but I’m just not so sure. Statement 2 looks really bad for him. If we can connect those dots for y and z, and we already have x, we should have all variables converted to numbers. Literally it all adds up. But I feel like I’m missing something. I can definitely get the sum of the first 4 terms and of the first 8 terms and of the first 12 terms; those are 10, and 20, and 30. But what about the number 98?

And that’s where Sarah Koenig’s trademark thoughtfulness-over-opinionatedry comes in. There is a giant hole in “Answer choice C’s case” against this problem. You can get the sequence in blocks of 4, but 98 is two past the last multiple of 4 (which is 96). The 97th term is easy: that’s x = 5. But the 98th term is tricky: it’s y, and we don’t know y unless we have z with it ( we just have the sum of the two). So we can’t solve for the 98th term. The answer has to be E – we just don’t know.

Now if you’ve heard yesterday’s episode, think about Dana’s “think of all the things that would have to have gone wrong, all the bad luck” rundown. “He lent his car and his phone to the guy who pointed the finger at him. That sucks for him. On the day that his girlfriend went missing. That’s awful luck…” And in real life she may be right – that’s a lot of probability to overcome. But on the GMAT they hand pick the questions. On this problem you can solve for the 97th term (up to 96 there are just blocks of 4 terms, and you know that each block sums to 10, and the 97th term is known as 5) or the 99th term (same thing, but add the sum of the 98th and 99th terms which you know is 2). But the GMAT hand-selected the tricky question just like Koenig hand-selected the Adnan Syed case for its mystery. GMAT Data Sufficiency questions are like Serial…it pays to be skeptical as you examine the evidence. It pays to think like Sarah Koenig. Unlike Jay, the statements will always be true and they’ll always be consistent, but like Serial in general you’ll sometimes find that you just don’t have enough information to definitively answer the question on everyone’s lips. So do your journalistic due diligence and look for alternative explanations (Don did it!). Next thing you know you’ll be “Stepping Out!!!” of the test center with a high GMAT score.

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Are you studying for the GMAT? We have free online GMAT seminars running all the time. And, be sure to find us on Facebook and Google+, and follow us on Twitter!

*By Brian Galvin*

1. **Burnout is real.** Throughout high school and my first two years of college, I filled my schedule with extracurriculars, classes, and travel, reasoning that I’d only have one high school and one undergraduate experience, and that I should try to get as much out of both as I could. However, at the end of my sophomore year, I realized that I wasn’t enjoying my work. My grades, my health, and my mood took a few heavy hits before I realized that the problem wasn’t that my major and my work didn’t fit my interests; I was just completely exhausted, mentally and physically, and the only possible solution was rest. Burnout isn’t a sign of laziness or weakness, but an occasionally unavoidable symptom of overwork. Unfortunately, if not dealt with in a healthy way, burnout can sap energy and waste time, and at worst can interrupt academic ambitions or make retention and learning impossible. I know plenty of people who became burned out earlier or later than I did in their academic careers, and productively used time off—often in the form of a gap year—to refresh themselves and to rekindle their passion for their work.

2. **Once you start school, it’s hard to stop.** A summer abroad in England and a light course load in my fall semester helped me recover from my burnout, but the process was slow and frustrating because I never found time to completely relax. My heavy reliance on financial aid and scholarships, the yearlong lease on my apartment, and my desire to graduate at the same time as my peers made the prospect of taking a semester off seem impossible, and my four-year class plan couldn’t easily be reorganized since the classes I needed to graduate were only offered in particular semesters. I chose to lighten rather than to pause my work, but academic and financial limits prevented me from lightening it as much as I would have liked to. By starting college, I had laid a fairly inflexible groundwork for the next four years, and two years later I had little choice but to stick to it, regardless of the fact that a break would probably have been extremely healthy at the time for me, both academically and personally.

3. **Once you stop school, it’s hard to start again.** Many students who take a gap year never return to their studies. College is demanding, expensive, and very tempting to postpone indefinitely once you’ve shifted away from the “school mode” mindset. Consider how committed you are to getting a college degree before deciding to take time off.

4. **If you’re going to take a gap year, do something meaningful with it.** Many students take a year to travel to incredible places or to gain valuable work experience, and return to school with amazing stories to tell and a unique perspective that ends up enriching their education. The ones I know recount these experiences as easily the most valuable part of their gap year. Others take a much-needed break, or save up money in order to be able to spend more time studying and less time working once they do enter college. College is fantastic, but so is the rest of the world, and both places have a lot to offer. The last thing you want to do is feel like you’ve done nothing but waste time while others around you have grown and learned.

6. **Admissions offices don’t discriminate one way or the other.** Again, the key is not where you spent your time but how you’ve used it. The things that appear on your resume and in your essays represent your interests and your character, whether they took place in a classroom or on a mission trip to Ecuador. A good friend of mine, who sat on an admissions committee for many years, once told me that some of the most interesting candidates he ever interviewed made themselves stand out by using a gap year to acquire unique experience.

There is no one perfect path. Just as everyone’s college experience is unique, every gap year is unique as well. If you’re really not sure, check with your university of interest to see what their policies are about taking time off. Three short years after starting college, I’ve already had to begin considering whether I’ll take time off before graduate school. Twelve months isn’t very long, just like four years (shockingly) isn’t very long; and ultimately, whether or not you take a gap year after high school won’t determine whether or not you succeed in life, whether you’ll be happy, or even whether you finish your education.

Plan on taking the SAT soon? We run a free online SAT prep seminar every few weeks. And, be sure to find us on Facebook and Google+, and follow us on Twitter!

*Courtney Tran is a student at UC Berkeley, studying Political Economy and Rhetoric. In high school, she was named a National Merit Finalist and National AP Scholar, and she represented her district two years in a row in Public Forum Debate at the National Forensics League National Tournament.*

One reason people spend a lot of time on these questions is that they try to read the entire passage thoroughly. This is normal because this is how most reading is done, be it in newspapers or periodicals or novels. However, on the GMAT, speed is the name of the game. If I were doing a book report on Shakespeare’s works, then I would read the text multiple times, looking for nuance and symbolism. The goal on the GMAT is quite different: you have roughly eight minutes to read a passage and then answer four questions about it. That isn’t much time, but it can work if you’re question-focused.

Why be question focused? (Rhetorical question) A typical passage may have 300-400 words, and you could be asked 20-30 different questions about the information contained within it. In reality, you will only be asked 3-4-5 questions about this text, so becoming an expert on the minutiae contained within seems like a complete waste of time. In fact, considering that you only have ~2 minutes per question, it is not only a waste of time but a distraction that will waste precious time and lower your score. The vast majority of questions will require you to go back to the passage and reread the relevant portion, so your initial read is only there to give you a general sense of the text. After the initial read, you should be able to answer broad, universal questions. However, for questions that deal in specifics, you’ll have to go back to the text.

Specific questions deal with (drum roll please) specific elements of the passage. At first glance, you wouldn’t necessarily recall such minute details, but if you know where to go back in the text, it becomes a trivial case of rereading until you find it. As an example, you might not remember what Luke Skywalker was wearing on Tatooine when he first meets Obi-Wan Kenobi, but you could just rewatch the first act of Star Wars and see for yourself. There is no need to memorize every minor detail, as long as you know where to find the answer, you can just look it up.

Let’s look at a GMAT passage and answer a question that deals with a specific element of the passage (note: this is the same passage I used in October for a function question).

Nearly all the workers of the Lowell textile mills of Massachusetts were unmarried daughters from farm families. Some of the workers were as young as ten. Since many people in the 1820s were disturbed by the idea of working females, the company provided well-kept dormitories and boarding-houses. The meals were decent and church attendance was mandatory. Compared to other factories of the time, the Lowell mills were clean and safe, and there was even a journal, The Lowell Offering, which contained poems and other material written by the workers, and which became known beyond New England. Ironically, it was at the Lowell Mills that dissatisfaction with working conditions brought about the first organization of working women.

The mills were highly mechanized, and were in fact considered a model of efficiency by others in the textile industry. The work was difficult, however, and the high level of standardization made it tedious. When wages were cut, the workers organized the Factory Girls Association. 15,000 women decided to “turn out”, or walk off the job. The Offering, meant as a pleasant creative outlet, gave the women a voice that could be heard by sympathetic people elsewhere in the country, and even in Europe. However, the ability of the women to demand changes was severely circumscribed by an inability to go for long without wages with which to support themselves and help support their families. The same limitation hampered the effectiveness of the Lowell Female Labor Reform Association (LFLRA), organized in 1844.

No specific reform can be directly attributed to the Lowell workers, but their legacy is unquestionable. The LFLRA’s founder, Sarah Bagley, became a national figure, testifying before the Massachusetts House of Representatives. When the New England Labor Reform League was formed, three of the eight board members were women. Other mill workers took note of the Lowell strikes, and were successful in getting better pay, shorter hours, and safer working conditions. Even some existing child labor laws can be traced back to efforts first set in motion by the Lowell Mill Women.

According to the passage, which of the following contributed to the inability of the workers at Lowell to have their demands met?

(A) The very young age of some of the workers made political organization impractical.

(B) Social attitudes of the time pressured women into not making demands.

(C) The Lowell Female Labor Reform Association was not organized until 1844.

(D) Their families depended on the workers to send some of their wages home.

(E) The people who were most sympathetic to the workers lived outside of New England.

If you’ve been following the Veritas technique on Reading Comprehension, then you should have spent about two minutes reading through the passage and summarizing each paragraph in a couple of words. If you didn’t do this, feel free to go back and do it now. Once completed, your summaries of each paragraph should be something like:

1) Lowell Mills and context

2) Labor strife and consequences

3) Legacy of Lowell Mills

Your exact wording may vary, but you want to keep it at about 3-5 words or so. This should give enough of a framework so you know where to go in every question. If we look at the question at hand, it asks why were the workers at Lowell unable to have their demands met. This has to be in the second paragraph, as that was the part that dealt with the actual worker strife.

Rereading this paragraph, we go through a description of what prompted the strike and then how many people participated. Directly following this is the line: “However, the ability of the women to demand changes was severely circumscribed by an inability to go for long without wages with which to support themselves and help support their families”. This was their downfall: they needed money to support themselves and their loved ones (unsurprisingly the downfall of most strikes). The wording used may be somewhat obtuse, but the context makes it quite clear that the issue was money. Going through the answer choices, D is the only option that is remotely close to what we want, and is therefore the correct answer.

On Reading Comprehension questions, it’s very easy to experience information overload (TL;DR for the new generation). A lot of information is contained in each passage, and this is not an accident. The test is designed to try and waste your time with frivolous sentences, so your goal is to read for overarching intent and know that you’ll have to revisit the text on most questions. Specific questions tend to ask about something minor, or possibly tangential, and therefore usually require you to reread the passage. Practice Reading Comprehension timing and you will find that you can answer these specific questions faster.

Plan on taking the GMAT soon? We have GMAT prep courses starting all the time. And, be sure to find us on Facebook and Google+, and follow us on Twitter!

*Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you weekly advice for success on your exam. After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.*

**1. CNN Student News (http://www.cnn.com/studentnews/)** – Hate reading the newspaper? Want to watch a fast-paced video instead? CNN Student News is your answer. One caveat: while the “fun facts” may interest you generally, they probably aren’t good material for the SAT essay. After all, how would you sculpt an argument about the importance of privacy based on the fact that dark chocolate is good for your heart? (No kidding, that was part of a video three months ago.)

**2. Sparknotes Videos** (**http://www.sparknotes.com/sparknotes/video/****) – **Pretty self-explanatory, Sparknotes Videos translate lengthy plot summaries into animated 7-10 minute videos that sum up several commonly-studied novels. While you’re watching, be sure to have your SAT2400 Workbook handy so that you can note down *full names of characters, the setting, relevant plot points, the author, *and* the time period. *

**3. History Class – **Why not kill two birds with one stone and refresh your memory on something (a battle, a major treaty, etc.) you recently had to study for a history test? While choosing events – and this goes for your current events and literary examples as well – always make sure that the example can be “twisted” in multiple fashions to fit multiple topics. Several such multi-faceted examples will ensure that you are prepared to answer any question with ease.

**4. Wikipedia** – Believe it or not, several students each class ask me if “it is okay” to use Wikipedia to look up specifics of their examples. Yes! This is not a high school research paper where your teacher has banned Wikipedia as a source. The website is a time-efficient way of noting the details you need for several examples in each category.

**5. Facebook News** – I mention this one because most of you are probably spending hours a day on Facebook anyways. You might even be inadvertently aware of what’s going on in current events because you just can’t help but glance to that upper right part of your home page! Let’s face it, taking a break from the “OMG LOOK AT THE SANDWICH I JUST ATE” pics will not only help you on the SAT but will prevent brain decay. Seriously. I studied Neuroscience in college.

Lastly, NO NO NO personal examples! Do as much as you can to make yourself seem more educated than the average test-taker. Be sure to keep in mind academic tone when you write your essay. For more guidance on how to master this, check out last week’s tip here. Happy Studying!

Plan on taking the SAT soon? We run a free online SAT prep seminar every few weeks. And, be sure to find us on Facebook and Google+, and follow us on Twitter!

*Sidhi Gosain studied Neuroscience at Brown University. She tutored for BRYTE, a student-led program which pairs undergraduate tutors with refugees from Southeast Asia, Africa, and the Middle East. While she is currently pursuing a film career in both acting and production in Los Angeles, Sidhi is excited to keep tutoring students to beat standardized tests!*

Most candidates will simply blindly ask a superior at their company for a recommendation with little to no connection or background on the candidates life plans and career goals. The best recommenders can speak confidently about your unique contributions in the workplace and how these experiences position you as a strong candidate.

The best way to begin starts with the selection process. Identify people in your life that can speak to how you operate in professional settings; this can include people from the workplace, civic or volunteer organizations, and even school. You want to select the people who know you best and can speak to your strengths in a positive and comprehensive way.

Once you identify these 3 or 4 potential recommenders start to build these relationships. I feel the best way to do this is through consistent personal interactions. In person is ideal via lunches and coffee chats but email updates and phone calls are an adequate virtual substitute if the potential recommender is not local. This should not feel like work, it is simply you connecting and cultivating a pre-existing relationship. As you build this relationship pick the recommender’s brain for advice on career related things to make them feel more invested in your future. With the relationship flourishing your request for them to write your recommendation will come as no surprise so utilize this improved relationship to better align on recommendation expectations. Writing recommendations is a huge undertaking for an already busy senior professional so the more vested they are in your success the better you can expect your recommendation to be.

Another key aspect of developing these relationships is continuing to excel in the capacity in which they know you. Whether it’s professionally, academically or civically this is not the time to slack. Use your performance as the real catalyst to build your relationship and get the recommendation you deserve.

Want to craft a strong application? Call us at 1-800-925-7737 and speak with an MBA admissions expert today. Click here to take our Free MBA Admissions Profile Evaluation! As always, be sure to find us on Facebook and Google+, and follow us on Twitter!

*Dozie A. is a Veritas Prep Head Consultant for the Kellogg School of Management at Northwestern University. His specialties include consulting, marketing, and low GPA/GMAT applicants.*

Let’s take a look at each individual essay:

Essay 1 – Resilience. Perseverance. Grit. Call it what you will…. Challenges can build character. Describe a challenging experience you’ve had. How were you tested? What did you learn?

The three adjectives signal right away what Kellogg wants from you in this essay. It’s all about self-reflection and maturity. So open up! It is not enough to simply offer up a challenging experience for this essay. Admissions is also looking for your thought process during this situation. Where was the strife? What made the situation so challenging? How did you overcome this challenging experience? How did this experience impact you moving forward? Be introspective and dive deep into the specific of the situation. Breakthrough essays will put the reader right in the middle of the conflict early on and show NOT tell the specific steps the applicant took to overcome the challenge while including the corresponding thought process during this experience. Also, since essay 2 is focused on a professional experience, this essay may be an obvious opportunity to get a little personal with your choice of topic.

Essay 2 – Leadership requires an ability to collaborate with and motivate others. Describe a professional experience that required you to influence people. What did this experience teach you about working with others, and how will it make you a better leader?

The core of the Kellogg MBA is development of interpersonal skills through collaborative learning. The school has always had the reputation as the premier teamwork school in the world. One rarely discussed nuance of the Kellogg MBA is that the school is not simply interested in developing team players but instead they want to develop leaders of teams. As you identify which experience makes sense here, select one that you can really tell a full and comprehensive story for. Too many candidates select anecdotes with limited scope, which really restricts the depth with which candidates can write.

A key trap many applicants fall into here is to not fully answer the question. The essay is about two things: leadership and influence; not addressing both will be a major missed opportunity to show off those fancy interpersonal skills Kellogg loves so much. The “influence” component of this essay is the more common area where candidate underwhelm. Breakthrough candidates will showcase how they have changed mindsets as leaders and the mechanisms by which they have successfully done so.

As with all Kellogg application components it is important to remain self-reflective while integrating the personal elements of your professional decision making into responses to these essays. Remain focused on these key tips and come decision day you will be seen as a breakthrough candidate by admissions.

Want to craft a strong application? Call us at 1-800-925-7737 and speak with an MBA admissions expert today. Click here to take our Free MBA Admissions Profile Evaluation! As always, be sure to find us on Facebook and Google+, and follow us on Twitter!

Last Digit of Base:

0 – Last digit of expression with any power will be 0.

1 – Last digit of expression with any power will be 1.

2 – 2, 4, 8, 6, 2, 4, 8, 6… Cyclicity is 4.

3 – 3, 9, 7, 1, 3, 9, 7, 1… Cyclicity is 4.

4 – 4, 6, 4, 6, 4, 6, 4, 6… Cyclicity is 2.

5 - Last digit of expression with any power will be 5.

6 – Last digit of expression with any power will be 6.

7 – 7, 9, 3, 1, 7, 9, 3, 1… Cyclicity is 4.

8 – 8, 4, 2, 6, 8, 4, 2, 6… Cyclicity is 4.

9 – 9, 1, 9, 1, 9, 1, 9, 1… Cyclicity is 2.

Cyclicity is nothing but pattern recognition. You see that when you multiply 2 by itself, there is a pattern of last digit which goes 2, 4, 8, 6, 2, 4, 8, 6 and so on. We can use the same principle for when a question asks us for the last two digits of the expression. Let me remind you first that here at QWQW, we sometimes flirt with the lines that define GMAT scope. Obviously, we do point out whenever we are indulging and that’s exactly what we are going to do in this post. We are carrying on for the love of Math and the Q51 score.

The last two digits of the base decide the last two digits of the expression. For example,

**Example 1: **Let’s look at powers of 11.

11^1 = 11

11^2 = 121

11^3 = 1331

11^4 = …41 (we should just multiply the last two digits together and ignore the rest)

11^5 = …51

11^6 = …61

11^7 = …71

Note that the last two digits are displaying a pattern depending on the power. So we expect the cyclicity here to be 10.

11^8 = …81

11^9 = …91

11^(10) = …01

11^(11) = …11

11^(12) = …21

and so on. So the last two digits should go from 11, 21 to 91, 01 and then go to 11 again. The cycle of 10 starts from power of 1, 11, 21 etc. This means that 11^(46) should have last two digits as 61, 11^(92) should have last two digits as 21 and 11^(168) should have last two digits as 81.

Let’s look at some other numbers now:

**Example 2:** Say, we need the last two digits of 6^{58}

6^1 = 6 (No second last digit)

6^2 = 36

6^3 = 216

6^4 = …96 (Just multiply the last two digits)

6^5 = …76

6^6 = …56

6^7 = …36

and hence starts the cycle again:

3, 1, 9, 7, 5, 3, 1, 9, 7, 5 and so on.

The new cycle with tens digit of 3 begins at the powers of 2, 7, 12, 17, 22, 27 etc. So the new cycle will also begin at power of 57 and 6^58 will have 1 as the tens digit.

**Example 3:** How about the last two digits of 7^102?

7^1 = 7 (No second last digit)

7^2 = 49

7^3 = 343

7^4 = …01

7^5 = …07

7^6 = …49

7^7 = …43

We see a cyclicity of 4 here: 49, 43, 01, 07, 49, 43, 01, 07 … and so on. The new cycle begins at 2, 6, 10, 14 i.e. even powers which are not multiples of 4. So a new cycle will begin at 102 too. So the last two digits of 7^(102) will be 49.

Now there can be many variations in the questions asking us to find the last two digits. We will use different concepts for different question types. Today we saw how to use pattern recognition. We will look at some other methods next week.

*Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the **GMAT** for Veritas Prep and regularly participates in content development projects such as this blog!*

However…

There are three knee-jerk questions that you should plug (if not chug) into your brain to ask yourself every time you see a Min/Max problem before you ask that fourth question “What’s my strategy?”:

- Do the numbers have to be integers?
- Is zero a possible value?
- Are repeat numbers possible?

In the Veritas Prep Word Problems lesson we refer to these problems as “scenario-driven” Min/Max problems precisely because of the above questions. The scenario created by the problem drives the whole thing, related mainly to those three above questions. Consider these four prompts and ask yourself which ones can definitively be answered:

*#1: “Four friends go fishing and catch a total of 10 fish. How many fish did the friend with the highest total catch?”*

*#2: “Four friends go fishing and catch a total of 10 fish. If no two friends caught the same number of fish, how many fish did the friend with the highest total catch?”*

*#3: “Four friends go fishing and catch a total of 10 fish. If each friend caught at least one fish but no two friends caught the same number, how many fish did the friend with the highest total catch?”*

*#4: “Four friends go fishing and catch a total of 10 pounds of fish. If each friend caught at least one fish but no two friends caught the same number, how many pounds of fish did the friend with the highest total catch?”*

Hopefully you can see the progression as this set builds. In the first problem, there’s clearly no way to tell. Did one friend catch all ten? Did everyone catch at least two and two friends tied with 3? You just don’t know. But then it gets interesting, based on the questions you need to ask yourself on all of these.

With #2, two big restrictions are in play. Fish must be integers, so you’re only dealing with the 11 integers 0 through 10. And if no two friends caught the same number there’s a limited number of unique values that can add up to 10. But the catch on this one should be evident after you’ve read #3. Zero *IS* possible in this case, so while the totals could be 1, 2, 3, and 4 (guaranteeing the answer of 4), if the lowest person could have caught 0 (that’s where “min/max” comes in – to maximize the top value you want to minimize the other values) there’s also the possibility for 0, 1, 2, and 7. Because the zero possibility was still lurking out there, there’s not quite enough information to solve this one. And that’s why you always have to ask yourself “is 0 possible?”.

#3 should showcase that. If 0 is no longer a possibility *AND* the numbers have to be integers *AND* the numbers can’t repeat, then the only option is 1 (the new min value since 0 is gone), 2 (because you can’t match 1), 3, and 4. The highest total is 4.

And #4 shows why the seemingly-irrelevant backstory of “friends going fishing” is so important. Pounds of fish can be nonintegers, but fish themselves have to be integers. So even though this prompt looks very similar to #3, because we’re no longer limited to integers it’s very easy for the values to not repeat and still give wildly different max values (1, 2, 3, and 4 or 1.5, 2, 3, and 3.5 for example).

As you can see, the scenario really drives the answer, although the fourth question “What is my strategy?” will almost always require some real work. Let’s take a look at a couple questions from the Veritas Prep Question Bank to illustrate.

**Question 1:**

Four workers from an international charity were selling shirts at a local event yesterday. Did one of the workers sell at least three shirts yesterday at the event?

(1) Together they sold 8 shirts yesterday at the event.

(2) No two workers sold the same number of shirts.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked

(C) Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient

(D) EACH statement ALONE is sufficient to answer the question asked

(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed

Before you begin strategizing, ask yourself the three major questions:

1) Do the values have to be integers? YES – that’s why the problem chose shirts.

2) Is zero possible? YES – it’s not prohibited, so that means you have to consider zero as a min value.

3) Can the numbers repeat? That’s why statement 2 is there. With the given information and with statement 1, numbers can repeat. That allows you to come up with the setup 2, 2, 2, and 2 for statement 1 (giving the answer “NO”) or 1, 2, 2, and 3 (giving the answer “YES” and proving this insufficient).

But when statement 2 says on its own that, NO, the numbers cannot repeat, that’s a much more impactful statement than most test-takers realize. Taking statement 2 alone, you have four integers that cannot repeat (and cannot be negative), so the smallest setup you can find is 0, 1, 2, and 3 – and with that someone definitely sold at least three shirts. Statement 2 is sufficient with really no calculations whatsoever, but with careful attention to the ever-important questions.

**Question 2:**

Last year, Company X paid out a total of $1,050,000 in salaries to its 21 employees. If no employee earned a salary that is more than 20% greater than any other employee, what is the lowest possible salary that any one employee earned?

(A) $40,000

(B) $41,667

(C) $42,000

(D) $50,000

(E) $60,000

Here ask yourself the same questions:

1) The numbers do not have to be integers.

2) Zero is theoretically possible (but probably constrained by the 20% difference restriction)

3) Numbers absolutely can repeat (which will be very important)

4) What’s your strategy? If you want the LOWEST possible single salary, then use your answer to #3 (they can repeat) and give the other 20 salaries the maximum. That way your calculation looks like:

x + 20(1.2x) = 1,050,000

Which breaks out to 25x = 1,050,000, and x = 42000. And notice how important the answer to #3 was – by knowing that numbers could repeat, you were able to quickly put together a smart strategy to minimize one single value.

The larger lesson is crucial here, though – these problems are often (but not always) fairly basic mathematically, but derive their difficulty from a situation that limits some options or allows for more than you’d think via integer restrictions, the possibility of zero, and the possibility of repeat values. Ask yourself these four questions, and your answer to the first three especially will maximize your efficiency on the strategic portion of the problem.

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*By Brian Galvin*