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Is a*b*c divisible by 24? (1) a,b, and c are consecutive

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Is a*b*c divisible by 24? (1) a,b, and c are consecutive [#permalink]  12 Dec 2010, 19:06
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Is a*b*c divisible by 24?

(1) a, b, and c are consecutive even integers
(2) a*b is divisible by 12

Please provide your thoughts on this one. The question comes from one of the GMAT Club Tests on Number Props.

[Reveal] Spoiler:
Here's the answer from the GMAT Club Test:

Statement (1) by itself is sufficient. One of any three consecutive even integers is divisible by 3. Because this integer is even, it is also divisible by 6. When multiplied by two more even integers, it renders a product that is divisible by 24.

Statement (2) by itself is insufficient. We need to know something about c.

Correct me if I'm wrong, but I thought Zero is not divisible 24? If either a, b, or c is zero, then a*b*c= 0, which is not divisible
[Reveal] Spoiler: OA
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Re: Data Suff - Number Property... [#permalink]  12 Dec 2010, 20:00
tarn151 wrote:
Please provide your thoughts on this one. The question comes from one of the GMAT Club Tests on Number Props.

Is a*b*c divisible by 24?

1) a,b, and c are consecutive even integers
2) a*b is divisible by 12

[Reveal] Spoiler:
Here's the answer from the GMAT Club Test:

Statement (1) by itself is sufficient. One of any three consecutive even integers is divisible by 3. Because this integer is even, it is also divisible by 6. When multiplied by two more even integers, it renders a product that is divisible by 24.

Statement (2) by itself is insufficient. We need to know something about c.

Correct me if I'm wrong, but I thought Zero is not divisible 24? If either a, b, or c is zero, then a*b*c= 0, which is not divisible

The answer is A. Why?

First, you need to factor out the number 24. 24 is composed of 4*6 or 2*2*2*3. So, in order for a*b*c to be divisible by 24, it must contain at least 3 2's and a 3 as its factors.

1) a, b and c are consecutive positive integers. This means two things: a, b and c have at least 3 2's, because they are all even, and one of them must be divisible by 3. Product of any 3 consecutive integers will be divisible by 3. Try it: 1,2,3 are divisible by 3. 4,5,6 are divisible by 3. 8,10,12 are divisible by 3. In addition, it is good to know that any set of 3 consecutive integers will be divisible by 3 or 3! Also, set of 8 consecutive integers will be divisible by 8! Any set of n consecutive integers will be divisible by n!
So, since a, b, and c contain at least 2*2*2*3, it must be divisible by 24.

2) If ab is divisible by 12, we know nothing about c. If c is 1, a*b*c is not divisible by 24. If c is 2, it is. So the answer is: maybe. Not sufficient.

Finals answer: A.

Hope this helps, friend.
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Re: Data Suff - Number Property... [#permalink]  13 Dec 2010, 01:09
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tarn151 wrote:
Please provide your thoughts on this one. The question comes from one of the GMAT Club Tests on Number Props.

Is a*b*c divisible by 24?

1) a,b, and c are consecutive even integers
2) a*b is divisible by 12

[Reveal] Spoiler:
Here's the answer from the GMAT Club Test:

Statement (1) by itself is sufficient. One of any three consecutive even integers is divisible by 3. Because this integer is even, it is also divisible by 6. When multiplied by two more even integers, it renders a product that is divisible by 24.

Statement (2) by itself is insufficient. We need to know something about c.

Correct me if I'm wrong, but I thought Zero is not divisible 24? If either a, b, or c is zero, then a*b*c= 0, which is not divisible

Note that an integer $$a$$ is a multiple of an integer $$b$$ (integer $$a$$ is a divisible by an integer $$b$$) means that $$\frac{a}{b}=integer$$: so, as 0 divided by any integer (except zero itself) yields an integer then yes, zero is a multiple of every integer (except zero itself).

Also on GMAT when we are told that $$a$$ is divisible by $$b$$ (or which is the same: "$$a$$ is multiple of $$b$$", or "$$b$$ is a factor of $$a$$"), we can say that:
1. $$a$$ is an integer;
2. $$b$$ is an integer;
3. $$\frac{a}{b}=integer$$.

BACK TO THE ORIGINAL QUESTION:
Is a*b*c divisible by 24?

(1) a, b, and c are consecutive even integers --> $$a=2k-2$$, $$b=2k$$ and $$c=2k+2$$ for some integer $$k$$ --> $$abc=(2k-2)2k(2k+2)=8(k-1)k(k+1)$$, now $$(k-1)$$, $$k$$, $$(k+1)$$ are 3 consecutive integers, which means that one of them must be a multiple of 3, thus $$abc$$ is divisible by both 8 and 3, so by 24. Sufficient.

Or even without the formulas: th product of 3 consecutive even integers will have 2*2*2=8 as a factor, plus out of 3 consecutive even integers one must be a multiple of 3, thus abc is divisible by both 8 and 3, so by 24.

(2) a*b is divisible by 12, clearly insufficient as no info about c (if ab=12 and c=1 answer will be NO but if ab=24 and c=any integer then the answer will be YES).

Answer: A.

Hope it's clear.
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Re: Data Suff - Number Property... [#permalink]  13 Dec 2010, 05:19
Thanks for the explanation.
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Re: Data Suff - Number Property... [#permalink]  13 Dec 2010, 21:39
Expert's post
tarn151 wrote:
Please provide your thoughts on this one. The question comes from one of the GMAT Club Tests on Number Props.

Is a*b*c divisible by 24?

1) a,b, and c are consecutive even integers
2) a*b is divisible by 12

[Reveal] Spoiler:
Here's the answer from the GMAT Club Test:

Statement (1) by itself is sufficient. One of any three consecutive even integers is divisible by 3. Because this integer is even, it is also divisible by 6. When multiplied by two more even integers, it renders a product that is divisible by 24.

Statement (2) by itself is insufficient. We need to know something about c.

Correct me if I'm wrong, but I thought Zero is not divisible 24? If either a, b, or c is zero, then a*b*c= 0, which is not divisible

The product of two consecutive integers will be divisible by 2. e.g. 3*4 or 1*2 or (-5)(-4) etc because one of the numbers will always be even.

The product of 3 consecutive integers will be divisible by 3 because there will be a multiple of 3 in 3 consecutive numbers e.g. 2*3*4 or 7*8*9 etc

The product of 4 consecutive integers will be divisible by 4 because there will be a multiple of 4 in 4 consecutive numbers e.g. 1*2*3*4 or 8*9*10*11 etc

and so on....

Stmnt 1: a,b, and c are consecutive even integers

Since a, b and c are even, each one of them has a 2 to give us an 8. Also, if we take their 2s out, we are left with 3 consecutive integers which will definitely have a multiple of 3 e.g. 6*8*10 gives us 2*2*2*(3*4*5).
Hence 3 consecutive even integers' product is divisible by 8 and by 3. Hence it is divisible by 24. Sufficient.

Stmnt 2: a*b is divisible by 12
Remember, this statement does not give any relation between a, b and c. Do not assume here that they are still even consecutive integers. If a*b is divisible by 12, it doesn't say anything about c. Also, we don't know if a*b is itself divisible by 24. Hence not sufficient.

Answer (A).
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Re: Data Suff - Number Property... [#permalink]  15 Jul 2011, 03:06
Cant the three numbers be
a = -2, b= 0, c = 2.
The question does not say the numbers have to be positive?
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Re: Data Suff - Number Property... [#permalink]  15 Jul 2011, 03:14
Cant the three numbers be
a = -2, b= 0, c = 2.
The question does not say the numbers have to be positive or non-zero.
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Re: Data Suff - Number Property... [#permalink]  15 Jul 2011, 09:22
Expert's post
eshabhide wrote:
Cant the three numbers be
a = -2, b= 0, c = 2.
The question does not say the numbers have to be positive or non-zero.

Yes, they could be, but that doesn't change the answer here. Zero is divisible by every positive integer.
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Re: Data Suff - Number Property... [#permalink]  04 Sep 2011, 22:32
Quote:
Please provide your thoughts on this one. The question comes from one of the GMAT Club Tests on Number Props.

Is a*b*c divisible by 24?

1) a,b, and c are consecutive even integers
2) a*b is divisible by 12

[Reveal] Spoiler:
[Reveal] Spoiler: OA

From statement 1
Prime factors of 24: 2,2,2,3
Consecutive even integers that results in 24 when multiplied together: 2, 4, 6
--> sufficient

From statement 2
a*b is a multiple of 12
a*b can be: 12, 24, 36, 48, etc...
--> insufficient

Answer: A
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Last edited by gmatopoeia on 05 Sep 2011, 23:24, edited 1 time in total.
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Re: Data Suff - Number Property... [#permalink]  05 Sep 2011, 22:25
+1 for A
Plug-in Numbers 2,4,6 for "1"
We dont the value of "C" so it is not sufficient.
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Re: Data Suff - Number Property...   [#permalink] 05 Sep 2011, 22:25
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