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Manager  Joined: 19 Nov 2007
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The number 523abc is divisible by 7,8,9. Then what is the  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 29% (02:54) correct 71% (02:33) wrong based on 224 sessions

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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

Originally posted by jade3 on 06 Nov 2009, 03:41.
Last edited by Bunuel on 09 Jul 2013, 10:44, edited 1 time in total.
Renamed the topic and edited the question.
Math Expert V
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The number 523abc is divisible by 7,8,9. Then what is the  [#permalink]

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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 cannot be more than 2.

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

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8
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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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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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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Bunuel wrote:
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.

I thought that for a number to be divisible by 8, its last three digits had to be divisible by 8, any clue on why this is not working in answer choice D?

Cheers!
J Veritas Prep GMAT Instructor V
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Posts: 10008
Location: Pune, India

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jlgdr wrote:
Bunuel wrote:
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.

I thought that for a number to be divisible by 8, its last three digits had to be divisible by 8, any clue on why this is not working in answer choice D?

Cheers!
J The options give us the value of a*b*c, not abc. If we get abc as 656, it is divisible by 8. The options give us a*b*c = 6*5*6 = 180. The product may or may not be divisible by 8.
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Thanks Karishma, anyways what I tried here was prime factorization to see which answer choice could be multiples of 9, that is having the whole sum of digits to be a multiple of 9 and only C and D worked. Now, it seems a bit complicated to decide between the two. I have both even too. Any clues on how to proceed from here? I still haven't used the fact that they need to be divisible by 7 and 8 but I'm not sure how to approach these tests quickly

Cheers
J
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jlgdr wrote:
Thanks Karishma, anyways what I tried here was prime factorization to see which answer choice could be multiples of 9, that is having the whole sum of digits to be a multiple of 9 and only C and D worked. Now, it seems a bit complicated to decide between the two. I have both even too. Any clues on how to proceed from here? I still haven't used the fact that they need to be divisible by 7 and 8 but I'm not sure how to approach these tests quickly

Cheers
J

523abc is divisible by 9 which means 5+2+3+a+b+c is divisible by 9 i.e. 1+a+b+c is divisible by 9.

When you prime factorize each option, you are left only with (D)

A. $$504 = 2^3 * 3^2 * 7$$
Since a, b and c must be single digit numbers, the only way you can assign values to them is 8, 9 and 7. 1+8+9+7 = 25 which is not divisible by 9.

B. $$532 = 2^2*7*19$$
19 cannot be assigned to any of a, b and c and hence this product is not possible.

C. $$210 = 2^2 * 3 * 5 * 7$$
You cannot combine them in any way such that you get three single digit numbers. Note that if 5 and 7 are multiplied by even 2, they will become double digit hence two of a, b and c must take values of 5 and 7. The third cannot be the leftover 12.

D. $$180 = 2^2 * 3^2 * 5$$
There are two options: 4, 9, 5 and 6, 6, 5.
6, 6, 5 satisfies since 1+6+6+5 = 18 divisible by 9

E. $$280 = 2^3 * 5 * 7$$
The digits must be 8, 5 and 7. But 1+8+5+7 is not divisible by 9.

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Bunuel wrote:
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.

I have a doubt Bunuel. Wouldn't that be equal to aba? because i tried solving it backwards using options. The only way option D could've factorized into single digits was 4,5,9 and for this case the singular digits would add up to 5+2+3+4+5+9 = 28 which would make it indivisible by 9. i understand it could have been 6,6,5 too but wouldn't that make it aba instead of abc?
Math Expert V
Joined: 02 Sep 2009
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gauravkaushik8591 wrote:
Bunuel wrote:
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.

I have a doubt Bunuel. Wouldn't that be equal to aba? because i tried solving it backwards using options. The only way option D could've factorized into single digits was 4,5,9 and for this case the singular digits would add up to 5+2+3+4+5+9 = 28 which would make it indivisible by 9. i understand it could have been 6,6,5 too but wouldn't that make it aba instead of abc?

I think you assume that a, b and c must be distinct digits. But this is not true: unless it is explicitly stated otherwise, different variables CAN represent the same number.

Hope it's clear.
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VeritasPrepKarishma wrote:
jlgdr wrote:
Thanks Karishma, anyways what I tried here was prime factorization to see which answer choice could be multiples of 9, that is having the whole sum of digits to be a multiple of 9 and only C and D worked. Now, it seems a bit complicated to decide between the two. I have both even too. Any clues on how to proceed from here? I still haven't used the fact that they need to be divisible by 7 and 8 but I'm not sure how to approach these tests quickly

Cheers
J

523abc is divisible by 9 which means 5+2+3+a+b+c is divisible by 9 i.e. 1+a+b+c is divisible by 9.

When you prime factorize each option, you are left only with (D)

A. $$504 = 2^3 * 3^2 * 7$$
Since a, b and c must be single digit numbers, the only way you can assign values to them is 8, 9 and 7. 1+8+9+7 = 25 which is not divisible by 9.

B. $$532 = 2^2*7*19$$
19 cannot be assigned to any of a, b and c and hence this product is not possible.

C. $$210 = 2^2 * 3 * 5 * 7$$
You cannot combine them in any way such that you get three single digit numbers. Note that if 5 and 7 are multiplied by even 2, they will become double digit hence two of a, b and c must take values of 5 and 7. The third cannot be the leftover 12.

D. $$180 = 2^2 * 3^2 * 5$$
There are two options: 4, 9, 5 and 6, 6, 5.
6, 6, 5 satisfies since 1+6+6+5 = 18 divisible by 9

E. $$280 = 2^3 * 5 * 7$$
The digits must be 8, 5 and 7. But 1+8+5+7 is not divisible by 9.

I find this method to be very time consuming. SVP  Status: The Best Or Nothing
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Re: The number 523abc is divisible by 7,8,9. Then what is the  [#permalink]

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LCM of 7,8 & 9 = 7 * 8 * 9 = 504

Whatever the number 523abc should be divisible by 504

We know that 504000 is divisible by 504

Adding multiples of 504 to 504000 to reach 523abc

We get two numbers:

523152 & 523656 (both are divisible by 504)

Product of abc = 10 OR 180

Option D fits in

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Re: The number 523abc is divisible by 7,8,9. Then what is the  [#permalink]

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523abc is divisible by 504.

To find abc first lets divide 523000 by 504

523000/504 = 137 and remainder is 352. -------------(1)

352 + 152 = 504
i.e. when 152 is added with the remainder 352 of (1) then in (1) quotient = 138 and remainder will be 0

523152 / 504 = 138 and remainder is 0
As per the question "abc" is a*b*c = 180
but here 1*5*2=10

523656/504 = 139 and remainder is 0
a*b*c= 6*5*6 = 180

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Swami.
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GRE 1: Q169 V154 Re: The number 523abc is divisible by 7,8,9. Then what is the  [#permalink]

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PareshGmat wrote:
LCM of 7,8 & 9 = 7 * 8 * 9 = 504

Whatever the number 523abc should be divisible by 504

We know that 504000 is divisible by 504

Adding multiples of 504 to 504000 to reach 523abc

We get two numbers:

523152 & 523656 (both are divisible by 504)

Product of abc = 10 OR 180

Option D fits in

What ?
Okay so you are telling me to find this hepttyy looking divisibility computation..
Looks Wrong to me..
but the method is tooooooooooo long
P.S => add a few more ooooo in tooooo
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