• NEW! FREE Beat The GMAT Quizzes
Hundreds of Questions Highly Detailed Reporting Expert Explanations
• 7 CATs FREE!
If you earn 100 Forum Points

Engage in the Beat The GMAT forums to earn
100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## There are 27 different three-digit integers that can be tagged by: AAPL ##### This topic has 5 expert replies and 0 member replies ### Top Member ## There are 27 different three-digit integers that can be ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult GMAT Prep There are 27 different three-digit integers that can be formed using only the digits 1, 2, and 3. If all 27 of the integers were listed, what would their sum be? A. 2,704 B. 2,990 C. 5,404 D. 5,444 E. 5,994 OA E. ### GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1449 messages Followed by: 32 members Upvotes: 59 AAPL wrote: GMAT Prep There are 27 different three-digit integers that can be formed using only the digits 1, 2, and 3. If all 27 of the integers were listed, what would their sum be? A. 2,704 B. 2,990 C. 5,404 D. 5,444 E. 5,994 $? = 111 + 112 + 113 + 121 + 122 + 123 + \ldots + 331 + 332 + 333$ $1 \,\, \, \underline {} \, \, \, \underline {} :\,111 + 112 + 113 + 121 + 122 + 123 + 131 + 132 + 133$ $\left\{ \begin{gathered} \,1 \to {\text{9}}\,\,{\text{times}}\,\,{\text{in}}\,\,{\text{hundreds}}\,\,{\text{digit}} \hfill \\ \,1,2,3 \to {\text{3}}\,\,{\text{times}}\,\,{\text{each}}\,\,{\text{in}}\,\,{\text{tens}}\,\,{\text{digit}} \hfill \\ \,1,2,3 \to {\text{3}}\,\,{\text{times}}\,\,{\text{each}}\,\,{\text{in}}\,\,{\text{units}}\,\,{\text{digit}} \hfill \\ \end{gathered} \right.$ ${\text{Same}}\,\,{\text{occurs}}\,\,{\text{with}}\,\,2 \,\,\, \underline {} \,\,\, \underline {} \, \,\,\, {\text{and}} \,\,\, \,\,3 \,\,\, \underline {} \,\,\, \underline {} \,\,,\,\,{\text{hence:}}$ ${\text{?}}\,\,\,{\text{:}}\,\,\,\left\{ \begin{gathered} \,1,2,3 \to {\text{9}}\,\,{\text{times}}\,\,{\text{in}}\,\,{\text{hundreds}}\,\,{\text{digit}} \hfill \\ \,1,2,3 \to {\text{3 + 3 + 3}}\,\,{\text{times}}\,\,{\text{each}}\,\,{\text{in}}\,\,{\text{tens}}\,\,{\text{digit}} \hfill \\ \,1,2,3 \to {\text{3 + 3 + 3}}\,\,{\text{times}}\,\,{\text{each}}\,\,{\text{in}}\,\,{\text{units}}\,\,{\text{digit}} \hfill \\ \end{gathered} \right.$ $? = 9\left( {1 + 2 + 3} \right) \cdot 100 + 9\left( {1 + 2 + 3} \right) \cdot 10 + 9\left( {1 + 2 + 3} \right)$ $? = 9 \cdot 6 \cdot \left( {100+10+1} \right) = 5994$ This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator ( Math for the GMAT) English-speakers :: https://www.gmath.net Portuguese-speakers :: https://www.gmath.com.br ### GMAT/MBA Expert GMAT Instructor Joined 25 May 2010 Posted: 15261 messages Followed by: 1862 members Upvotes: 13060 GMAT Score: 790 AAPL wrote: GMAT Prep There are 27 different three-digit integers that can be formed using only the digits 1, 2, and 3. If all 27 of the integers were listed, what would their sum be? A. 2,704 B. 2,990 C. 5,404 D. 5,444 E. 5,994 Given any set that is symmetrical about the median: Sum = (quantity)(median) The set of 3-digit integers that can be formed from the digits 1,2 and 3 is symmetrical about the median (222): ...212, 213, 221, 222, 223, 231, 232... Thus: Sum of the 27 integers = (quantity)(median) = 27*222 = 5994. The correct answer is E. Another approach: There are 3 positions in each integer: hundreds place, tens place, units place. Each digit will appear in each position 27/3 = 9 times. Thus, in each position, there will be nine 1's, nine 2's, and nine 3's. Sum of the digits in each position = 9(1+2+3) = 54. Sum for the hundreds place = 54*100 = 5400. Sum for the tens place = 54*10 = 540. Sum for the units place = 54*1 = 54. Sum of all the integers = 5400+540+54 = 5994. Similar problem: https://www.beatthegmat.com/we-make-4-digit-codes-and-each-digit-of-the-code-form-from-1-t288802.html _________________ Mitch Hunt Private Tutor for the GMAT and GRE GMATGuruNY@gmail.com If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon. Available for tutoring in NYC and long-distance. For more information, please email me at GMATGuruNY@gmail.com. Student Review #1 Student Review #2 Student Review #3 Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now. ### GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1449 messages Followed by: 32 members Upvotes: 59 GMATGuruNY wrote: Given any set that is symmetrical about the median: Sum = (quantity)(median) The set of 3-digit integers that can be formed from the digits 1,2 and 3 is symmetrical about the median (222): ...212, 213, 221, 222, 223, 231, 232... Thus: Sum of the 27 integers = (quantity)(median) = 27*222 = 5994. Outstanding, Mitch. Congrats! Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator ( Math for the GMAT) English-speakers :: https://www.gmath.net Portuguese-speakers :: https://www.gmath.com.br ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 12838 messages Followed by: 1247 members Upvotes: 5254 GMAT Score: 770 AAPL wrote: GMAT Prep There are 27 different three-digit integers that can be formed using only the digits 1, 2, and 3. If all 27 of the integers were listed, what would their sum be? A. 2,704 B. 2,990 C. 5,404 D. 5,444 E. 5,994 OA E. Yes, great work, Mitch! I thought I'd mention that, once we know the correct answer is a multiple of 222 (which is also a multiple of 3), then the correct answer will be a multiple of 3. Useful property: If a number is a multiple of 3, then the sum of its digits is also a multiple of 3. When we use this property to check the answer choices, we see that only E (5994) is a multiple of 3 (since 5 + 9 + 9 + 4 = 27, and 27 is a multiple of 3) Cheers, Brent _________________ Brent Hanneson â€“ Creator of GMATPrepNow.com Use our video course along with Sign up for our free Question of the Day emails And check out all of our free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMATâ€™s FREE 60-Day Study Guide and reach your target score in 2 months! ### GMAT/MBA Expert GMAT Instructor Joined 25 Apr 2015 Posted: 2456 messages Followed by: 18 members Upvotes: 43 AAPL wrote: GMAT Prep There are 27 different three-digit integers that can be formed using only the digits 1, 2, and 3. If all 27 of the integers were listed, what would their sum be? A. 2,704 B. 2,990 C. 5,404 D. 5,444 E. 5,994 Some of the integers that have this property are 123, 111, 213, and 322. Of course, itâ€™s possible to list all 27 such integers and then add them up. However, it will be too time-consuming. Therefore, we will use a shortcut by arguing that, of these 27 integers, the digits 1, 2, and 3 must appear in the hundreds position 9 times each. Using the same logic, they each will also appear in the tens position 9 times each and in the units position 9 times each. Thus, the sum of these integers is: (100 + 200 + 300) x 9 + (10 + 20 + 30) x 9 + (1 + 2 + 3) x 9 (600) x 9 + (60) x 9 + (6) x 9 (666) x 9 5,994 Answer: E _________________ Scott Woodbury-Stewart Founder and CEO scott@targettestprep.com See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews • Free Trial & Practice Exam BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • 5 Day FREE Trial Study Smarter, Not Harder Available with Beat the GMAT members only code • 5-Day Free Trial 5-day free, full-access trial TTP Quant Available with Beat the GMAT members only code • Get 300+ Practice Questions 25 Video lessons and 6 Webinars for FREE Available with Beat the GMAT members only code • Free Veritas GMAT Class Experience Lesson 1 Live Free Available with Beat the GMAT members only code • FREE GMAT Exam Know how you'd score today for$0

Available with Beat the GMAT members only code

• Magoosh
Study with Magoosh GMAT prep

Available with Beat the GMAT members only code

• 1 Hour Free
BEAT THE GMAT EXCLUSIVE

Available with Beat the GMAT members only code

• Free Practice Test & Review
How would you score if you took the GMAT

Available with Beat the GMAT members only code

• Award-winning private GMAT tutoring
Register now and save up to \$200

Available with Beat the GMAT members only code

### Top First Responders*

1 Jay@ManhattanReview 77 first replies
2 Brent@GMATPrepNow 66 first replies
3 GMATGuruNY 33 first replies
4 Ian Stewart 24 first replies
5 Scott@TargetTestPrep 16 first replies
* Only counts replies to topics started in last 30 days
See More Top Beat The GMAT Members

### Most Active Experts

1 Scott@TargetTestPrep

Target Test Prep

200 posts
2 Max@Math Revolution

Math Revolution

94 posts
3 Brent@GMATPrepNow

GMAT Prep Now Teacher

92 posts
4 Jay@ManhattanReview

Manhattan Review

83 posts
5 GMATGuruNY

The Princeton Review Teacher

57 posts
See More Top Beat The GMAT Experts