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What is the smallest five digit number that is divisible by 16, 24, 36

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What is the smallest five digit number that is divisible by 16, 24, 36 [#permalink]

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New post 19 Nov 2010, 10:35
00:00
A
B
C
D
E

Difficulty:

  75% (hard)

Question Stats:

42% (01:24) correct 58% (01:30) wrong based on 50 sessions

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What is the smallest five digit number that is divisible by 16, 24, 36 and 54.

A. 10320
B. 10080
C. 10032
D. 10368
E. None of the above
[Reveal] Spoiler: OA

Last edited by Bunuel on 12 Sep 2017, 02:47, edited 1 time in total.
Edited the OA.

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Re: What is the smallest five digit number that is divisible by 16, 24, 36 [#permalink]

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New post 19 Nov 2010, 10:58
nades09 wrote:
What is the smallest five digit number that is divisible by 16, 24, 36 and 54.

A. 10320
B. 10080
C. 10032
D. 10368
E. None of the above

I feel the answer should be D and not E. Please help

Thanks
NAD


What's the source of this question? It seems remarkably un-GMATesque, since you need to do a lot of crunching to come up with the right answer; especially given the inclusion of choice E, which makes backsolving almost impossible.

That said, let's do some crunching. When you see big numbers and factors or multiples, your first instinct should be to break things down into primes.

Let's break down our component numbers and create the lowest common multiple:

16 = 2*2*2*2

so, our LCM must have 2^4 in it

24 = 2*2*2*3

we already have lots of 2s, so we just need to bring in the 3. Our work in progress is now:

2^4 * 3

36 = 4*9 = 2*2*3*3

we already have lots of 2s and one 3, so we need to bring in one more 3. Our current LCM:

2^4 * 3^2

54 = 2*27 = 2*3*3*3

we need to add 1 more 3, giving us a final LCM of:

2^4 * 3^3

The question asks what's the smallest 5 digit number that's a multiple of our LCM. Here's where the question gets unfair, since there's no elegant (i.e. short and sweet) solution - brute force is now required, something that almost never happens on the GMAT.

Well, our number is 16*27 = 432

432 * 20 = 8640... that's a good starting point, let's work up from there:

8640 + 432 = 9072 + 432 = 9504 + 432 = 9936 + 432 = 10368... choose D.

So, not only is this a poorly constructed question, but (assuming that you copied it correctly) the answer provided is also wrong!

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Re: What is the smallest five digit number that is divisible by 16, 24, 36 [#permalink]

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New post 19 Nov 2010, 11:50
Thanks!
That is how I derived the answer, except instead of adding the LCM, I just factorised each option to select the one that contains 2^4 and 3^2. The question is from 4Gmat question set.

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Re: What is the smallest five digit number that is divisible by 16, 24, 36 [#permalink]

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New post 20 Nov 2010, 03:13
If question is correct, then asnwer choice is D. When I selected D, I got a red flag. Please check both.
Could any one post short cuts. I did conventional way...too much time consuming....

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Re: What is the smallest five digit number that is divisible by 16, 24, 36 [#permalink]

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New post 20 Nov 2010, 08:03
anandthiru wrote:
If question is correct, then asnwer choice is D. When I selected D, I got a red flag. Please check both.
Could any one post short cuts. I did conventional way...too much time consuming....


Because of choice (E), we can't use shortcuts and be sure that we're correct.

Let's look at the choices:

A. 10320
B. 10080
C. 10032
D. 10368
E. None of the above

Since 36 has to be a factor, we know that 9 also must be a factor. The quick way to check if 9 is a factor of a number is to sum the digits and see if you get a multiple of 9.

Based on that rule we can eliminate A and C, since the digits of both A and C sum to 6. However, we still have B (sum to 9) and D (sum to 18) in the running. E, of course, is also still possible.

Even if we had eliminated 3 of the first 4 choices, we'd have no quick way of knowing whether the remaining answer were the smallest 5 digit number that satisfies the criteria of the question.

Because there is no creative solution, this question would never appear on the actual GMAT, which wants to reward you for finding the best solution; questions that can only be solved via traditional methods are rare on the exam, and none of them require this level of calculation.

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Re: What is the smallest five digit number that is divisible by 16, 24, 36 [#permalink]

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New post 20 Nov 2010, 12:14
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anandthiru wrote:
If question is correct, then asnwer choice is D. When I selected D, I got a red flag. Please check both.
Could any one post short cuts. I did conventional way...too much time consuming....


As Stuart has very clearly explained above, to be divisible by 16, 24, 36 and 54, the number should be divisible by \(2^4\) and \(3^3\) so the number should be a multiple of 432.

Out of the given options, lets eliminate 10320 and 10032 right away because they are not even divisible by 9, forget about by 27.

Out of the remaining options, 10080 is not divisible by 27 (because when I divide 10080 by 9, I get 1120 as quotient which is not divisible by 3)
10368 is divisible by 27 (when I divide 10368 by 9, I get 1152 which is divisible by 3) and 368 is divisible by 16 (To check if a number is divisible by 16, you need to see if last 4 digits are divisible by 16)

It is clear then, that 10368 is the answer since it is a multiple of 432 and definitely the smallest 5 digit multiple (The multiple of 432 smaller than 10368 will be 4 digit because the difference between 10368 and that multiple will be 432)

This is an alternate solution. How much time it would save depends on whether you are comfortable with divisibility rules and multiplication tables. As Stuart pointed out, the calculations are tedious.
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Re: What is the smallest five digit number that is divisible by 16, 24, 36 [#permalink]

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New post 22 Nov 2010, 03:26
Karishma's method is smarter. I calculated the LCM (432), then divided 10000 by the LCM (432). Then subtracted the remainder(64) from LCM (432). Finally added the result (368) to 10000 and got the answer (10368). Took me ages!!

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Re: What is the smallest five digit number that is divisible by 16, 24, 36 [#permalink]

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Re: What is the smallest five digit number that is divisible by 16, 24, 36   [#permalink] 01 Oct 2017, 06:32
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