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# The number 523abc

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The number 523abc [#permalink]  06 Nov 2009, 03:41
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Question Stats:

40% (05:47) correct 60% (00:16) wrong based on 5 sessions
The number 523abc is divisible by 7,8,9. Then what is the value of a*b*c

A. 504
B. 532
C. 210
D. 180
E. 280
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Re: The number 523abc [#permalink]  06 Nov 2009, 05:45
I dont know if my method is the best, but i guess the ans is surely available and quicker as well. Here i go:

523abc is divisible by 7, 8 and 9. product of 7, 8 and 9 is 504 and hence 523abc must be divisible by 504 as well. Lets assume, abc are 000, dividing 523000 by 504, we get a remainder of 352, which means it is short by 152 (504-352) to be perfectly divisible by 504. so the number has to be 523152 or 523656 (which is 523152+504). so abc should be 1,5,2 or 6,5,6. Product of these sets of integers is, 10 or 180. Since 10 is not in the option, 180 should be the answer.

I would go with OA D.
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Re: The number 523abc [#permalink]  06 Nov 2009, 06:14
The number 523abc is divisible by 7,8,9. Then what is the value of a*b*c

A. 504
B. 532
C. 210
D. 180
E. 280

LCM of 7, 8 and 9 is 504, thus 523abc must be divisible by 504.

523abc=523000+abc
523000 divided by 504 gives a remainder of 352.
Hence, 352+abc=k*504.

k=1 abc=152 --> a*b*c=10
k=2 abc=656 --> a*b*c=180
As abc is three digit number k can not be more than 2.

Two answers? Well only one is listed in answer choices, so D.

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Re: The number 523abc [#permalink]  06 Nov 2009, 08:21
I also choose D.
a, b, c cannot be 0.
The number is divisible by 9 =>sum of 5+2+3+a+b+c must be divisible by 9, so a+b+c can be 8, 17 or 26 (as maximum sum of 3 1-digit number is 27).
It is also divisible by 8 => It will be divisible by 4 and 2 => bc must be divisible by 4 and c must be even. so c just can be 2 or 6.
It is also divisible by 7.
So abc must be 656, its product is 180.
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Re: The number 523abc [#permalink]  09 May 2011, 18:59
I broke the answer choices into their prime factors, then tested the different 3 digit combos ending in an even number for which one would make 5+2+3+a+b+c be divisible by 9. The only number with prime factors that does that is 180. Hence, the answer is D.
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Re: The number 523abc   [#permalink] 09 May 2011, 18:59
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