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

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VP
Joined: 22 Nov 2007
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If x represents the sum of all the positive three-digit [#permalink]

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09 Mar 2008, 21:23
This topic is locked. If you want to discuss this question please re-post it in the respective forum.

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?

3
6
11
22
222
SVP
Joined: 29 Aug 2007
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09 Mar 2008, 22:29
marcodonzelli 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?

3
6
11
22
222

all 3 digits +ve numbers using a, b and c:
abc
acb
bca
bac
cab
cba

So sum = 100 [2(a+b+c)] + 10 [2(a+b+c)] + [2 (a+b+c)]
sum = 2 (a+b+c) (100+10+1)
sum = 2 (a+b+c) (111)
sum = 222 (a+b+c)

so its E.
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CEO
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Location: New York City
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11 Mar 2008, 13:45
GMAT TIGER wrote:
marcodonzelli 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?

3
6
11
22
222

all 3 digits +ve numbers using a, b and c:
abc
acb
bca
bac
cab
cba

So sum = 100 [2(a+b+c)] + 10 [2(a+b+c)] + [2 (a+b+c)]
sum = 2 (a+b+c) (100+10+1)
sum = 2 (a+b+c) (111)
sum = 222 (a+b+c)

so its E.

GMATTIGER, i am not quite understanding the logic of your answer although the OA is E.

I understand x is an with digit values of 100 [2(a+b+c)] & 10 [2(a+b+c)] & [2 (a+b+c)]. Why do we factor out A+B+C and what does it represent?
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Director
Joined: 08 Jun 2007
Posts: 583
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11 Mar 2008, 16:48
marcodonzelli 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?

3
6
11
22
222

I knew the solution before..ie 222

3 digit number can be represented by
xyz = 100x + 10y + x
xzy = 100x + 10z + y
.............................
.............................
This translates into .
(xyz +...) = 222 ( x + y + z )
Re: digits - gmatprep   [#permalink] 11 Mar 2008, 16:48
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