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# If x is an integer, then x(x 1)(x k) must be evenly

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Director
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If x is an integer, then x(x 1)(x k) must be evenly [#permalink]  29 Jun 2008, 21:06
If x is an integer, then x(x – 1)(x – k) must be evenly divisible by three when k is any of the following values EXCEPT
-4
-2
-1
2
5

Any short cut, the OE is not helpful
Senior Manager
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Re: x(x – 1)(x – k) [#permalink]  29 Jun 2008, 21:13
I would go for B.

The product may not be divisible by 3 if x-k is not equal to x-2, x-5, x-(-1), x-(-4). Because x or x-1 may not be divisible by 3.
SVP
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Re: x(x – 1)(x – k) [#permalink]  29 Jun 2008, 21:22
gmatnub wrote:
If x is an integer, then x(x – 1)(x – k) must be evenly divisible by three when k is any of the following values EXCEPT
-4
-2
-1
2
5

Any short cut, the OE is not helpful

make or break the expression into a form such that it is either in a consecutive form or a multiple of 3 as under:
1. x(x – 1)(x + 4) = x(x – 1)(x + 1 +3) = x(x – 1)[(x + 1) +3]
2. x(x – 1)(x + 2)
3. x(x – 1)(x + 1)
4. x(x – 1)(x - 2) = x(x – 1)(x + 1 - 3)
5. x(x – 1)(x - 5) = x(x – 1)(x +1 - 6)

B. except -2, all could be k such that x(x – 1)(x - k) is divisible by 3. in all options except second, the integers are in eith consecutive order or in at least one of them is divisible by 3 but B is not at any case.
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Manager
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Re: x(x – 1)(x – k) [#permalink]  29 Jun 2008, 22:27
I say shortcut for such problems is always trying a couple of numbers and see which one satisfies.
for eg
Take x=5
Just plug in the options. B doesnt satisfy
CEO
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Schools: Chicago (Booth) - Class of 2011
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Re: x(x – 1)(x – k) [#permalink]  29 Jun 2008, 22:33
Expert's post
1. x(x – 1)(x – k) is divisible by 3 if x,x-1,x-k is consecutive integers: x(x – 1)(x – 2) for k=2
2. Now, subtraction or addition of 3 to any of members does not change divisibility: -7,-4,-1,2,5,8. -2 is out of the set.
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SVP
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Re: x(x – 1)(x – k) [#permalink]  30 Jun 2008, 10:45
i dont fully get it ... what if x=3 ? thats what i used, and it turned out that if you use that, then for all k values in the question, the product is divisible by 3 :S

I guess for these types of questions, you shouldnt use the value youre dividing by ?
CEO
Joined: 17 Nov 2007
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Concentration: Entrepreneurship, Other
Schools: Chicago (Booth) - Class of 2011
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Kudos [?]: 2247 [0], given: 359

Re: x(x – 1)(x – k) [#permalink]  30 Jun 2008, 10:49
Expert's post
pmenon wrote:
i dont fully get it ... what if x=3 ? thats what i used, and it turned out that if you use that, then for all k values in the question, the product is divisible by 3 :S

I guess for these types of questions, you shouldnt use the value youre dividing by ?

Reread the question - must be evenly divisible by three [at any x]

x=3 and k=any - works
x=5 and k=0 - does not work
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Director
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Re: x(x – 1)(x – k) [#permalink]  30 Jun 2008, 10:55
gmatnub wrote:
If x is an integer, then x(x – 1)(x – k) must be evenly divisible by three when k is any of the following values EXCEPT
-4
-2
-1
2
5

Any short cut, the OE is not helpful

my 5 sec solution. x*(x-1)*(x-2) is divisible by 3. x*(x-1)*(x-k) must be divisible by 3 if k = 2(mod 3)

notice that -4, -1, 2, 5 = 2(mod 3). -2 = 1(mod 3) -> B
SVP
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Re: x(x – 1)(x – k) [#permalink]  30 Jun 2008, 11:06
walker wrote:
pmenon wrote:
i dont fully get it ... what if x=3 ? thats what i used, and it turned out that if you use that, then for all k values in the question, the product is divisible by 3 :S

I guess for these types of questions, you shouldnt use the value youre dividing by ?

Reread the question - must be evenly divisible by three [at any x]

x=3 and k=any - works
x=5 and k=0 - does not work

thanks walker, but im still confused. i get the general concept, but having problems applying it. i assume evenly divisible means that the quotient is an even number ? how does this change things ? gah im so confused !
CEO
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Schools: Chicago (Booth) - Class of 2011
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Re: x(x – 1)(x – k) [#permalink]  30 Jun 2008, 11:28
Expert's post
pmenon wrote:
thanks walker, but im still confused. i get the general concept, but having problems applying it. i assume evenly divisible means that the quotient is an even number ? how does this change things ? gah im so confused !

the quotient x(x-1)(x-k) div 3 is always even number, because one of the number, x or (x-1), is always even.
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Re: x(x – 1)(x – k)   [#permalink] 30 Jun 2008, 11:28
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