Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack

 It is currently 28 May 2017, 13:32

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# If x represents the sum of all the positive three-digit

Author Message
TAGS:

### Hide Tags

Intern
Joined: 02 Oct 2009
Posts: 16
Followers: 0

Kudos [?]: 47 [5] , given: 5

If x represents the sum of all the positive three-digit [#permalink]

### Show Tags

13 Nov 2009, 20:35
5
KUDOS
28
This post was
BOOKMARKED
00:00

Difficulty:

95% (hard)

Question Stats:

46% (02:00) correct 54% (01:59) wrong based on 1006 sessions

### HideShow timer Statistics

If x represents the sum of all the positive three-digit 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
[Reveal] Spoiler: OA

Last edited by Bunuel on 30 Jul 2012, 04:29, edited 2 times in total.
Manager
Joined: 05 Jun 2009
Posts: 77
Followers: 1

Kudos [?]: 2 [0], given: 1

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?
Math Expert
Joined: 02 Sep 2009
Posts: 39037
Followers: 7750

Kudos [?]: 106470 [33] , given: 11626

Re: this is what it has come down to [#permalink]

### Show Tags

13 Nov 2009, 21:34
33
KUDOS
Expert's post
18
This post was
BOOKMARKED
rvthryet wrote:
If x represents the sum of all the positive three-digit 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...
[Reveal] Spoiler:
OA E

Using THREE non-zero 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 $$222$$.

_________________
Manager
Joined: 11 Sep 2009
Posts: 129
Followers: 6

Kudos [?]: 362 [0], given: 6

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 three-digit 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...
[Reveal] Spoiler:
OA E

Using THREE non-zero 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 $$222$$.

Good explanation, exactly how I solved it. I love questions with elegant solutions like this. +1
CEO
Joined: 17 Nov 2007
Posts: 3585
Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
GMAT 1: 750 Q50 V40
Followers: 575

Kudos [?]: 3984 [0], given: 360

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 must-have app especially if you aim at 700+ | PrepGame

Manager
Joined: 10 Aug 2009
Posts: 123
Followers: 3

Kudos [?]: 16 [1] , given: 13

### Show Tags

03 Mar 2010, 04:26
1
KUDOS
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: 22
Followers: 0

Kudos [?]: 5 [0], given: 0

Re: If x represents the sum of all the positive three-digit [#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: 118
Location: India
WE: Marketing (Manufacturing)
Followers: 1

Kudos [?]: 126 [0], given: 16

Re: If x represents the sum of all the positive three-digit [#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
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15501
Followers: 651

Kudos [?]: 210 [1] , given: 0

Re: If x represents the sum of all the positive three-digit [#permalink]

### Show Tags

19 Sep 2013, 10:20
1
KUDOS
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Manager
Joined: 23 May 2013
Posts: 126
Followers: 1

Kudos [?]: 62 [0], given: 110

Re: If x represents the sum of all the positive three-digit [#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: 77
Location: United States
GMAT 1: 590 Q41 V29
GMAT 2: 540 Q44 V20
GPA: 3.5
WE: Programming (Computer Software)
Followers: 0

Kudos [?]: 64 [0], given: 24

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 three-digit 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...
[Reveal] Spoiler:
OA E

Using THREE non-zero 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 $$222$$.

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: 39037
Followers: 7750

Kudos [?]: 106470 [0], given: 11626

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 three-digit 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...
[Reveal] Spoiler:
OA E

Using THREE non-zero 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 $$222$$.

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.
_________________
Manager
Joined: 29 Aug 2013
Posts: 77
Location: United States
GMAT 1: 590 Q41 V29
GMAT 2: 540 Q44 V20
GPA: 3.5
WE: Programming (Computer Software)
Followers: 0

Kudos [?]: 64 [0], given: 24

Re: If x represents the sum of all the positive three-digit [#permalink]

### Show Tags

02 Oct 2013, 03:27
Hi Bunuel,

Thanks for the clarification. I thought it is the sum of all such 3-digit numbers that have distinct numbers.

What in the case "x is the sum of all the 3-digit 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)
Followers: 0

Kudos [?]: 5 [0], given: 23

Re: If x represents the sum of all the positive three-digit [#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.
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15501
Followers: 651

Kudos [?]: 210 [0], given: 0

Re: If x represents the sum of all the positive three-digit [#permalink]

### Show Tags

18 Jan 2016, 18:30
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7380
Location: Pune, India
Followers: 2291

Kudos [?]: 15146 [1] , given: 224

Re: If x represents the sum of all the positive three-digit [#permalink]

### Show Tags

18 Jan 2016, 22:29
1
KUDOS
Expert's post
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
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15501
Followers: 651

Kudos [?]: 210 [0], given: 0

Re: If x represents the sum of all the positive three-digit [#permalink]

### Show Tags

06 Feb 2017, 11:57
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Director
Joined: 14 Nov 2016
Posts: 900
Location: Malaysia
Followers: 25

Kudos [?]: 465 [1] , given: 158

Re: If x represents the sum of all the positive three-digit [#permalink]

### Show Tags

31 Mar 2017, 00:56
1
KUDOS
rvthryet wrote:
If x represents the sum of all the positive three-digit 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, Three-Digit Divisibility).
Attachments

Untitled.jpg [ 64.06 KiB | Viewed 911 times ]

_________________

Be challenged at EVERY MOMENT.

Each stage of the journey is crucial to attaining new heights of knowledge.

Math Expert
Joined: 02 Sep 2009
Posts: 39037
Followers: 7750

Kudos [?]: 106470 [0], given: 11626

Re: If x represents the sum of all the positive three-digit [#permalink]

### Show Tags

31 Mar 2017, 01:50
ziyuen wrote:
rvthryet wrote:
If x represents the sum of all the positive three-digit 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, Three-Digit Divisibility).

Edited. Thank you.
_________________
Re: If x represents the sum of all the positive three-digit   [#permalink] 31 Mar 2017, 01:50
Similar topics Replies Last post
Similar
Topics:
77 Of the three-digit positive integers whose three digits are all differ 18 21 May 2017, 18:44
2 If for positive integer x, y the sum of all the digit of x is 170 and 10 27 Apr 2016, 01:35
2 If x represents the sum of the interior angles of a regular hexagon an 5 17 Dec 2014, 06:36
21 Let abc and dcb represent three-digit positive integers. 9 18 Mar 2017, 09:41
21 The three-digit positive integer x has the hundreds, 11 28 Apr 2017, 04:57
Display posts from previous: Sort by