is N divisible by 24 ?

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GMATMadeEasy: Legendary Member; Posts: 768; Joined: Mon Nov 30, 2009 3:46 am; Thanked: 21 times; Followed by:7 members

is N divisible by 24 ?

by GMATMadeEasy » Thu Sep 30, 2010 2:43 am

If X is even, and N = (X)(X+1)(X+2),
is N divisible by 24?

OA is YES

Could someone explain please the solution for it.

I agree X must have a 2, X+2 must have a 2, and a 3 in total because there are three consecutive intergs.
That makes 2*2*3 , divisible by 12 maximum.

Thanks for conceptual explanation please.

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Source: Beat The GMAT — Problem Solving | Thu Sep 30, 2010 2:43 am

ashokkadam: Master | Next Rank: 500 Posts; Posts: 210; Joined: Wed Apr 16, 2008 11:25 pm; Thanked: 6 times; Followed by:1 members

by ashokkadam » Thu Sep 30, 2010 2:56 am

Simply follow the below test cases....

X N= (X)(X+1)(X+2) Divisible by 24?
------------------------------------------------------------
2 2 * 3 * 4 = 24 Yes
-2 -2 * -1 * 0 = 0 Yes
4 4 * 5 * 6 = 120 Yes
-4 -4 * -3 * -2 = -24 Yes

So you see that N is always divisible by 24. Answer is Yes!! Hope that helps!!

GMATMadeEasy wrote:If X is even, and N = (X)(X+1)(X+2),
is N divisible by 24?

OA is YES

Could someone explain please the solution for it.

I agree X must have a 2, X+2 must have a 2, and a 3 in total because there are three consecutive intergs.
That makes 2*2*3 , divisible by 12 maximum.

Thanks for conceptual explanation please.

Force and mind are opposites; morality ends where a gun begins.

Quote

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pm; Location: New York, NY; Thanked: 13060 times; Followed by:1906 members; GMAT Score:790

by GMATGuruNY » Thu Sep 30, 2010 2:59 am

GMATMadeEasy wrote:If X is even, and N = (X)(X+1)(X+2),
is N divisible by 24?

OA is YES

Could someone explain please the solution for it.

I agree X must have a 2, X+2 must have a 2, and a 3 in total because there are three consecutive intergs.
That makes 2*2*3 , divisible by 12 maximum.

Thanks for conceptual explanation please.

24 = 2*3*4, so to be divisible by 24, N must be divisible by 2, 3 and 4.

If x=2, then N = 2(2+1)(2+2) = 2*3*4.
Since x must be a multiple of 2, N must be a multiple of 2*3*4, which means it must be divisible by 2, 3 and 4.
Thus, N is divisible by 24.

Last edited by GMATGuruNY on Thu Sep 30, 2010 3:48 am, edited 1 time in total.

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ankur.agrawal: Master | Next Rank: 500 Posts; Posts: 261; Joined: Wed Mar 31, 2010 8:37 pm; Location: Varanasi; Thanked: 11 times; Followed by:3 members

by ankur.agrawal » Thu Sep 30, 2010 3:14 am

I have a confusion here:

Sumwhere in the manhattan guides i have read that even Number '0' is considered even. If we apply this, does it makes a difference to this question?

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pm; Location: New York, NY; Thanked: 13060 times; Followed by:1906 members; GMAT Score:790

by GMATGuruNY » Thu Sep 30, 2010 3:19 am

ankur.agrawal wrote:I have a confusion here:

Sumwhere in the manhattan guides i have read that even Number '0' is considered even. If we apply this, does it makes a difference to this question?

Yes, 0 is even. It is also divisible by -- and thus a multiple of -- every other integer.

So if N is a multiple of 2*3*4, then N could be 0 (which is a multiple of every other integer) or a negative multiple of 2*3*4 = 24.

Last edited by GMATGuruNY on Thu Sep 30, 2010 3:24 am, edited 1 time in total.

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ashokkadam: Master | Next Rank: 500 Posts; Posts: 210; Joined: Wed Apr 16, 2008 11:25 pm; Thanked: 6 times; Followed by:1 members

by ashokkadam » Thu Sep 30, 2010 3:19 am

@ankur.agrawal '0' is indeed an even number, so you must test with this number too for this question.

ankur.agrawal wrote:I have a confusion here:

Sumwhere in the manhattan guides i have read that even Number '0' is considered even. If we apply this, does it makes a difference to this question?

Force and mind are opposites; morality ends where a gun begins.

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goyalsau: Legendary Member; Posts: 866; Joined: Mon Aug 02, 2010 6:46 pm; Location: Gwalior, India; Thanked: 31 times

by goyalsau » Thu Sep 30, 2010 4:42 am

GMATGuruNY wrote:

Yes, 0 is even. It is also divisible by -- and thus a multiple of -- every other integer.

Can you please explain it in detail.
As i understand 0 is even and a multiple of every other integer.

And i read some where is not positive but neither negative.

I am really confused with Zero.

What i understand is that it is a positive , a even integer and a multiple of every integer ( whether it is positive or negative )
In other way 0 is not a negative integer.

Please Guru Correct me if I am wrong in any way, I think this is a very serious concept problem and i as know Gmat love to test it.

I would like to ask one more than is 1 a odd integer.

And this is the link of a question that i got wrong because of this Zero Concept
https://www.beatthegmat.com/sequence-t66944.html#301225

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ashokkadam: Master | Next Rank: 500 Posts; Posts: 210; Joined: Wed Apr 16, 2008 11:25 pm; Thanked: 6 times; Followed by:1 members

by ashokkadam » Thu Sep 30, 2010 4:47 am

Just remember this:
0 is neither positive nor negative(i.e. non-positive and non-negative).
0 is even.

goyalsau wrote:
GMATGuruNY wrote:

Yes, 0 is even. It is also divisible by -- and thus a multiple of -- every other integer.

Can you please explain it in detail.
As i understand 0 is even and a multiple of every other integer.

And i read some where is not positive but neither negative.

I am really confused with Zero.

What i understand is that it is a positive , a even integer and a multiple of every integer ( whether it is positive or negative )
In other way 0 is not a negative integer.

Please Guru Correct me if I am wrong in any way, I think this is a very serious concept problem and i as know Gmat love to test it.

I would like to ask one more than is 1 a odd integer.

And this is the link of a question that i got wrong because of this Zero Concept
https://www.beatthegmat.com/sequence-t66944.html#301225

Force and mind are opposites; morality ends where a gun begins.

Quote

GMATMadeEasy: Legendary Member; Posts: 768; Joined: Mon Nov 30, 2009 3:46 am; Thanked: 21 times; Followed by:7 members

by GMATMadeEasy » Thu Sep 30, 2010 6:24 am

If x=2, then N = 2(2+1)(2+2) = 2*3*4.
Since x must be a multiple of 2, N must be a multiple of 2*3*4, which means it must be divisible by 2, 3 and 4.

This is where I have the issue .

X is even so we know X has one 2 and we know X+2 has to one 2 or two 2s
X has two 2s as factors (or divisible by 4)

basically, I try to put different approaches , things start to look convoluted.

Do we have authentic way of solving it by figuring out factors (2*2*2*3) in the expression given.

Let's ignore the issue of 0. For those who are curious, 0 is neither positive nor negative but is an even integer. . use this definition and should be on your way. From OG .

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 am; Location: Montreal; Thanked: 1090 times; Followed by:355 members; GMAT Score:780

by Ian Stewart » Thu Sep 30, 2010 11:45 am

GMATGuruNY wrote: 24 = 2*3*4, so to be divisible by 24, N must be divisible by 2, 3 and 4.

If x=2, then N = 2(2+1)(2+2) = 2*3*4.
Since x must be a multiple of 2, N must be a multiple of 2*3*4, which means it must be divisible by 2, 3 and 4.
Thus, N is divisible by 24.

This isn't a legitimate proof. If you were instead asked "If x is divisible by 3, is x(x+1)(x+2) divisible by 60?" you'd certainly get the wrong answer if you simply tested the value x=3 and tried to generalize from there; you'd think the answer was yes, but the answer is 'not necessarily' (if x = 6, for example, x(x+1)(x+2) is not divisible by 5, and thus not divisible by 60).

In the original question above, if x is even, then x and x+2 are consecutive even integers. Every second even integer is a multiple of 4. So if you take any two consecutive even integers, exactly one of them will be divisible by 4. For that reason, in this question, x(x+2) must be a multiple of 2*4 = 8. Further, among any three consecutive integers, you always have exactly one multiple of 3, since multiples of 3 are three apart. So one of x, x+1 or x+2 is divisible by 3, and the product x(x+1)(x+2) is divisible by 3*8 = 24.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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ankur.agrawal: Master | Next Rank: 500 Posts; Posts: 261; Joined: Wed Mar 31, 2010 8:37 pm; Location: Varanasi; Thanked: 11 times; Followed by:3 members

by ankur.agrawal » Thu Sep 30, 2010 11:53 am

Ian Stewart wrote:
GMATGuruNY wrote: 24 = 2*3*4, so to be divisible by 24, N must be divisible by 2, 3 and 4.

If x=2, then N = 2(2+1)(2+2) = 2*3*4.
Since x must be a multiple of 2, N must be a multiple of 2*3*4, which means it must be divisible by 2, 3 and 4.
Thus, N is divisible by 24.
This isn't a legitimate proof. If you were instead asked "If x is divisible by 3, is x(x+1)(x+2) divisible by 60?" you'd certainly get the wrong answer if you simply tested the value x=3 and tried to generalize from there; you'd think the answer was yes, but the answer is 'not necessarily' (if x = 6, for example, x(x+1)(x+2) is not divisible by 5, and thus not divisible by 60).

In the original question above, if x is even, then x and x+2 are consecutive even integers. Every second even integer is a multiple of 4. So if you take any two consecutive even integers, exactly one of them will be divisible by 4. For that reason, in this question, x(x+2) must be a multiple of 2*4 = 8. Further, among any three consecutive integers, you always have exactly one multiple of 3, since multiples of 3 are three apart. So one of x, x+1 or x+2 is divisible by 3, and the product x(x+1)(x+2) is divisible by 3*8 = 24.

Dats the best approach IAN . U rock!!!

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pm; Location: New York, NY; Thanked: 13060 times; Followed by:1906 members; GMAT Score:790

by GMATGuruNY » Thu Sep 30, 2010 12:00 pm

Ian Stewart wrote:
GMATGuruNY wrote: 24 = 2*3*4, so to be divisible by 24, N must be divisible by 2, 3 and 4.

If x=2, then N = 2(2+1)(2+2) = 2*3*4.
Since x must be a multiple of 2, N must be a multiple of 2*3*4, which means it must be divisible by 2, 3 and 4.
Thus, N is divisible by 24.
This isn't a legitimate proof. If you were instead asked "If x is divisible by 3, is x(x+1)(x+2) divisible by 60?" you'd certainly get the wrong answer if you simply tested the value x=3 and tried to generalize from there; you'd think the answer was yes, but the answer is 'not necessarily' (if x = 6, for example, x(x+1)(x+2) is not divisible by 5, and thus not divisible by 60).

In the original question above, if x is even, then x and x+2 are consecutive even integers. Every second even integer is a multiple of 4. So if you take any two consecutive even integers, exactly one of them will be divisible by 4. For that reason, in this question, x(x+2) must be a multiple of 2*4 = 8. Further, among any three consecutive integers, you always have exactly one multiple of 3, since multiples of 3 are three apart. So one of x, x+1 or x+2 is divisible by 3, and the product x(x+1)(x+2) is divisible by 3*8 = 24.

I should have clarified my reasoning. Because x is even, N = even*odd*even (all consecutive), which means that N will always be a multiple of 2, 3 and 4, and hence a multiple of 24:

2*3*4
4*5*6
6*7*8
8*9*10

I personally find it easier to handle these sort of questions less conceptually and more concretely by examining real numbers. Looking at the list above, I can see that every threesome has a multiple of 2, a multiple of 3, and a multiple of 4, so the product must be divisible by 2*3*4 = 24.

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GMATMadeEasy: Legendary Member; Posts: 768; Joined: Mon Nov 30, 2009 3:46 am; Thanked: 21 times; Followed by:7 members

by GMATMadeEasy » Thu Sep 30, 2010 3:02 pm

@Ian Stewart : You fullflll my objective of posting the question, It is crystal clear now. A BIG thnks.

@GMATGURUNY : I am going to include your approach as backup approach but I enjoy algebericaly and nerdy approach