Mar 20 07:00 AM PDT  09:00 AM PDT Join a FREE 1day workshop and learn how to ace the GMAT while keeping your fulltime job. Limited for the first 99 registrants. Mar 19 08:00 AM PDT  09:00 AM PDT Beat the GMAT with a customized study plan based on your needs! Learn how to create your preparation timeline, what makes a good study plan and which tools you need to use to build the perfect plan. Register today! Mar 20 09:00 PM EDT  10:00 PM EDT Strategies and techniques for approaching featured GMAT topics. Wednesday, March 20th at 9 PM EDT Mar 23 07:00 AM PDT  09:00 AM PDT Christina scored 760 by having clear (ability) milestones and a trackable plan to achieve the same. Attend this webinar to learn how to build trackable milestones that leverage your strengths to help you get to your target GMAT score.
Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 02 Oct 2009
Posts: 16

If x represents the sum of all the positive threedigit
[#permalink]
Show Tags
Updated on: 30 Jul 2012, 04:29
Question Stats:
45% (02:10) correct 55% (02:18) wrong based on 1258 sessions
HideShow timer Statistics
If x represents the sum of all the positive threedigit numbers that can be constructed using each of the distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible? (A) 3 (B) 6 (C) 11 (D) 22 (E) 222
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by rvthryet on 13 Nov 2009, 20:35.
Last edited by Bunuel on 30 Jul 2012, 04:29, edited 2 times in total.
Added the OA




Math Expert
Joined: 02 Sep 2009
Posts: 53709

Re: this is what it has come down to
[#permalink]
Show Tags
13 Nov 2009, 21:34
rvthryet wrote: If x represents the sum of all the positive threedigit numbers that can be constructed using each of the distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible? (A) 3 (B) 6 (C) 11 (D) 22 (E) 222 I have never really understood the thinking behind this... Using THREE nonzero digits a,b,c only, we can construct 3!=6 numbers: abc, acb, bac, bca, cab, cba. Their sum will be: \(x=(100a+10b+c)+(100a+10c+b)+(100b+10a+c)+(100b+10c+a)+(100c+10a+b)+(100c+10b+a)=\) \(=200*(a+b+c)+20*(a+b+c)+2*(a+b+c)=\) \(=222*(a+b+c)\) Largest integer by which x MUST be divisible is \(222\). Answer: E (222).
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics




Manager
Joined: 05 Jun 2009
Posts: 72

Re: this is what it has come down to
[#permalink]
Show Tags
13 Nov 2009, 20:40
where did this question come from wow I have like no idea where to begin I would assume 123 and 987 which are two combinations are both both divisible by 3 as the GCD so 3? A?



Manager
Joined: 11 Sep 2009
Posts: 130

Re: this is what it has come down to
[#permalink]
Show Tags
13 Nov 2009, 21:47
Bunuel wrote: rvthryet wrote: If x represents the sum of all the positive threedigit numbers that can be constructed using each of the distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible? (A) 3 (B) 6 (C) 11 (D) 22 (E) 222 I have never really understood the thinking behind this... Using THREE nonzero digits a,b,c only, we can construct 3!=6 numbers: abc, acb, bac, bca, cab, cba. Their sum would be: \(x=(100a+10b+c)+(100a+10c+b)+(100b+10a+c)+(100b+10c+a)+(100c+10a+b)+(100c+10b+a)=\) \(=200*(a+b+c)+20*(a+b+c)+2*(a+b+c)=\) \(=222*(a+b+c)\) Largest integer by which x MUST be divisible is \(222\). Answer: E (222). Good explanation, exactly how I solved it. I love questions with elegant solutions like this. +1



CEO
Joined: 17 Nov 2007
Posts: 3420
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth)  Class of 2011

Re: this is what it has come down to
[#permalink]
Show Tags
13 Nov 2009, 21:59
We can also solve this one without math using symmetry: hundreds, tens and units are symmetric, so sum can be written as (y)*111. We need to check that y is even. For example, for fixed a at hundred position, there is two bc,cb combinations. Therefore, a is included twice (even number of times) into sum of hundreds. So, it is 222 By the way, it is the first time when I add something after Bunuel
_________________
HOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android)  The OFFICIAL GMAT CLUB PREP APP, a musthave app especially if you aim at 700+  Limited GMAT/GRE Math tutoring in Chicago



Manager
Joined: 10 Aug 2009
Posts: 116

Re: Testing number properties
[#permalink]
Show Tags
03 Mar 2010, 04:26
E
Maybe there is a faster way to do it but I did it like this:
How many ways can you arrange abc? abc acb bac bca cab cba
which are equivalent to: 100a + 10b + c 100a + 10c + b 100b + 10a + c 100b + 10c + a 100c + 10a + b 100c + 10b + a
if you add them all together you get 222a + 222b + 222c



Intern
Joined: 03 Dec 2010
Posts: 21

Re: If x represents the sum of all the positive threedigit
[#permalink]
Show Tags
31 Mar 2012, 03:27
To Bunuel, I've gone thorugh ur notes for each Quant topic and I try to solve topic wise questions from gmatclub. Sometimes I'm not able to figure out how to start with the problem, or I should say how to apply the properties learned since, the techniques you give in your solution for a given problem are not there in properties or formulaes. What do you recommend ? I plan to give my Gmat nxt mnth end. This Tuesday, Veritas prep test I took I scored 600, Q44, verbal 33. Kindly assist. Thanks.



Manager
Joined: 26 Jul 2011
Posts: 85
Location: India
WE: Marketing (Manufacturing)

Re: If x represents the sum of all the positive threedigit
[#permalink]
Show Tags
30 Jul 2012, 04:26
.Though I was able to solve it (in a random way), but was unable to come up with a concrete approach. @NickK kudos for that perfect one. This is how I did.....
The question asked for the largest divisor and thus we need to form 6 largest number that could be made using 3 distinct nonzero digits....987+978+897+879+798+789 = 5328...start from the largest number provided in the answer..222 divides 5328 completely hence is the answer



Manager
Joined: 23 May 2013
Posts: 99

Re: If x represents the sum of all the positive threedigit
[#permalink]
Show Tags
01 Oct 2013, 06:40
ratinarace wrote: .Though I was able to solve it (in a random way), but was unable to come up with a concrete approach. @NickK kudos for that perfect one. This is how I did.....
The question asked for the largest divisor and thus we need to form 6 largest number that could be made using 3 distinct nonzero digits....987+978+897+879+798+789 = 5328...start from the largest number provided in the answer..222 divides 5328 completely hence is the answer Agree, substitution works the best for 'must be true' problems.
_________________
“Confidence comes not from always being right but from not fearing to be wrong.”



Manager
Joined: 29 Aug 2013
Posts: 74
Location: United States
Concentration: Finance, International Business
GMAT 1: 590 Q41 V29 GMAT 2: 540 Q44 V20
GPA: 3.5
WE: Programming (Computer Software)

Re: this is what it has come down to
[#permalink]
Show Tags
02 Oct 2013, 00:27
Bunuel wrote: rvthryet wrote: If x represents the sum of all the positive threedigit numbers that can be constructed using each of the distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible? (A) 3 (B) 6 (C) 11 (D) 22 (E) 222 I have never really understood the thinking behind this... Using THREE nonzero digits a,b,c only, we can construct 3!=6 numbers: abc, acb, bac, bca, cab, cba. Their sum will be: \(x=(100a+10b+c)+(100a+10c+b)+(100b+10a+c)+(100b+10c+a)+(100c+10a+b)+(100c+10b+a)=\) \(=200*(a+b+c)+20*(a+b+c)+2*(a+b+c)=\) \(=222*(a+b+c)\) Largest integer by which x MUST be divisible is \(222\). Answer: E (222). Hi Bunuel, Can you please explain me what will be the value of "x" in this question. If it were asked what is the value of x? Thanks!



Math Expert
Joined: 02 Sep 2009
Posts: 53709

Re: this is what it has come down to
[#permalink]
Show Tags
02 Oct 2013, 03:12
shameekv wrote: Bunuel wrote: rvthryet wrote: If x represents the sum of all the positive threedigit numbers that can be constructed using each of the distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible? (A) 3 (B) 6 (C) 11 (D) 22 (E) 222 I have never really understood the thinking behind this... Using THREE nonzero digits a,b,c only, we can construct 3!=6 numbers: abc, acb, bac, bca, cab, cba. Their sum will be: \(x=(100a+10b+c)+(100a+10c+b)+(100b+10a+c)+(100b+10c+a)+(100c+10a+b)+(100c+10b+a)=\) \(=200*(a+b+c)+20*(a+b+c)+2*(a+b+c)=\) \(=222*(a+b+c)\) Largest integer by which x MUST be divisible is \(222\). Answer: E (222). Hi Bunuel, Can you please explain me what will be the value of "x" in this question. If it were asked what is the value of x? Thanks! We cannot say what x is. If a, b, and c, are 1, 2, and 3 respectively, then x = 123 + 132 + 213 + 231 + 312 + 321 = 1,332 = 6*222 (the least possible value of x). ... If a, b, and c, are 7, 8, and 9 respectively, then x = 789 + 798 + 879 + 897 + 978 + 987 = 5,328 = 24*222 (the greatest possible value of x). Hope it helps.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Manager
Joined: 29 Aug 2013
Posts: 74
Location: United States
Concentration: Finance, International Business
GMAT 1: 590 Q41 V29 GMAT 2: 540 Q44 V20
GPA: 3.5
WE: Programming (Computer Software)

Re: If x represents the sum of all the positive threedigit
[#permalink]
Show Tags
02 Oct 2013, 03:27
Hi Bunuel,
Thanks for the clarification. I thought it is the sum of all such 3digit numbers that have distinct numbers.
What in the case "x is the sum of all the 3digit numbers that have distinct numbers". How do you calculate the value of x in such case. I tried many things but couldn't work it out.
I saw such type of question recently where x was required to be calculated but the digits could be repeated and that made it simple. But I couldn't figure out with this restriction. Could you please help me out on that?
Thanks, Shameek



Intern
Joined: 08 Oct 2012
Posts: 1
Location: United States
Concentration: General Management, Technology
GPA: 2.3
WE: Engineering (Computer Software)

Re: If x represents the sum of all the positive threedigit
[#permalink]
Show Tags
26 Oct 2014, 12:52
Shamee, to solve the problem in a simpler manner why don't you assume the numbers a, b and c to be 1, 2 and 3 respectively?
Thus, the distinct numbers that can be formed would be  123 132 213 231 312 321
If you sum these up you get a total of 1332.
Then proceed to plug in the answer options to find the greatest number that divides 1332.
From the options  (A) 3  Yes (B) 6  Yes (C) 11  No (D) 22  No (E) 222  Yes
Clearly, since 222 is the greatest, E is the right option.



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8988
Location: Pune, India

Re: If x represents the sum of all the positive threedigit
[#permalink]
Show Tags
18 Jan 2016, 22:29
pritishpratap wrote: Shamee, to solve the problem in a simpler manner why don't you assume the numbers a, b and c to be 1, 2 and 3 respectively?
Thus, the distinct numbers that can be formed would be  123 132 213 231 312 321
If you sum these up you get a total of 1332.
Then proceed to plug in the answer options to find the greatest number that divides 1332.
From the options  (A) 3  Yes (B) 6  Yes (C) 11  No (D) 22  No (E) 222  Yes
Clearly, since 222 is the greatest, E is the right option. Here is the catch in "assuming values" in this question: The question is a "must be true" question. How do you know that what holds for values 1, 2 and 3 will be true for values say 2, 3 and 7 too? What if sum of numbers formed by 2, 3 and 7 is not divisible by 222? You do need to apply logic to confirm "must be true".
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >



Senior SC Moderator
Joined: 14 Nov 2016
Posts: 1319
Location: Malaysia

Re: If x represents the sum of all the positive threedigit
[#permalink]
Show Tags
31 Mar 2017, 00:56
rvthryet wrote: If x represents the sum of all the positive threedigit numbers that can be constructed using each of the distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?
(A) 3 (B) 6 (C) 11 (D) 22 (E) 222 Bunuel, This question has been wrongly tagged. The original source is Manhattan Prep, Challenge Problems (2002, December 2, ThreeDigit Divisibility).
Attachments
Untitled.jpg [ 64.06 KiB  Viewed 9819 times ]
_________________
"Be challenged at EVERY MOMENT."
“Strength doesn’t come from what you can do. It comes from overcoming the things you once thought you couldn’t.”
"Each stage of the journey is crucial to attaining new heights of knowledge."
Rules for posting in verbal forum  Please DO NOT post short answer in your post!
Advanced Search : https://gmatclub.com/forum/advancedsearch/



Math Expert
Joined: 02 Sep 2009
Posts: 53709

Re: If x represents the sum of all the positive threedigit
[#permalink]
Show Tags
31 Mar 2017, 01:50
ziyuen wrote: rvthryet wrote: If x represents the sum of all the positive threedigit numbers that can be constructed using each of the distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?
(A) 3 (B) 6 (C) 11 (D) 22 (E) 222 Bunuel, This question has been wrongly tagged. The original source is Manhattan Prep, Challenge Problems (2002, December 2, ThreeDigit Divisibility). Edited. Thank you.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Manager
Joined: 30 Dec 2016
Posts: 235
GPA: 4
WE: Business Development (Other)

If x represents the sum of all the positive threedigit
[#permalink]
Show Tags
09 Oct 2017, 03:10
I solved it in a bit different way. Not sure if this is correct. There are 3 numbers a, b, c so there can be 6 arrangements of these numbers. So, 6 possible numbers are there ( just to be sure i am not missing) abc +acb +bac +bca +cba +cab6a+6b+6cNow if we factor out 6 > 6( a+b+c ) from this we know the answer must be a multiple of 6. It can not be 6 as a+b+c wpuld yield some integer and 6*someinteger > 6 . So the only possible outcome is 222 which is a multiple of 6 other than Choice B. Answer E
_________________
Regards SandySilva
____________ Please appreciate the efforts by pressing +1 KUDOS (:



Manager
Joined: 02 Jul 2016
Posts: 110

Re: If x represents the sum of all the positive threedigit
[#permalink]
Show Tags
22 Aug 2018, 06:30
Bunuel wrote: rvthryet wrote: If x represents the sum of all the positive threedigit numbers that can be constructed using each of the distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible? (A) 3 (B) 6 (C) 11 (D) 22 (E) 222 I have never really understood the thinking behind this... Using THREE nonzero digits a,b,c only, we can construct 3!=6 numbers: abc, acb, bac, bca, cab, cba. Their sum will be: \(x=(100a+10b+c)+(100a+10c+b)+(100b+10a+c)+(100b+10c+a)+(100c+10a+b)+(100c+10b+a)=\) \(=200*(a+b+c)+20*(a+b+c)+2*(a+b+c)=\) \(=222*(a+b+c)\) Largest integer by which x MUST be divisible is \(222\). Answer: E (222). I tried solving it using the formula (n1)!*(sum of the digits)*(111…..n times) n =3 sum of the digits = a+b+c so (31)! *(a+b+c)*111 222*(a+b+c) Hence clearly the number will be divisible by 222




Re: If x represents the sum of all the positive threedigit
[#permalink]
22 Aug 2018, 06:30






