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# If x is a positive integer, is x^3-3x^2+2x divisible by 4?

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If x is a positive integer, is x^3-3x^2+2x divisible by 4? [#permalink]

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06 Feb 2012, 13:30
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If x is a positive integer, is x^3-3x^2+2x divisible by 4?

(1) x=4y+4, where y is an integer
(2) x=2z+2, where z is an integer

[Reveal] Spoiler:
in the above (1) is exactly a multiple for 4, so sufficient BUT in (2) if z = 0 , then x = 2, which when substituted in quest will not be divisible by 4. right?
but the answer is D, both alone sufieicient

can anyone guide. do i have a wrong approach towards DS??
[Reveal] Spoiler: OA
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06 Feb 2012, 13:44
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kashishh wrote:
if x is a +ve integer, is x³ - 3x² + 2x divisible by 4?
(1) x = 4y + 4, where y is an integer
(2) x = 2z + 2, where z is an integer

in the above (1) is exactly a multiple for 4, so sufficient BUT in (2) if z = 0 , then x = 2, which when substituted in quest will not be divisible by 4. right?
but the answer is D, both alone sufieicient

can anyone guide. do i have a wrong approach towards DS??

If x is a positive integer , is x^3 - 3x^2+2x divisible by 4?

(1) x=4y+4, where y is an integer --> since x itself is divisible by 4 then x^3-3x^2+2x is divisible by 4. Sufficient.

(2) x=2z+2, where z is an integer --> x^3-3x^2+2x=x(x^2-3x+2)=(2z+2)(4z^2+8z+4-6z-6+2)=4(z+1)(2z^2+z) --> hence this expression is divisible by 4. Sufficient.

As for your question: if x=2 then x^3-3x^2+2x=0. Now, zero is divisible by EVERY integer except zero itself, as 0/integer=integer.

For more on this topic check Number Theory chapter of Math Book: math-number-theory-88376.html

Hope it's clear.
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Re: If x is a positive integer , is x^3 - 3x^2+2x divisible by 4 [#permalink]

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20 Feb 2013, 21:38
What level would this be considered? 650?
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Re: If x is a positive integer , is x^3 - 3x^2+2x divisible by 4 [#permalink]

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21 Feb 2013, 03:30
hitman5532 wrote:
What level would this be considered? 650?

Yes, I'd say the difficulty level is ~650.
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Re: If x is a positive integer, is x^3-3x^2+2x divisible by 4? [#permalink]

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26 Feb 2013, 20:16
If x is a positive integer, is x^3-3x^2+2x divisible by 4?

(1) x=4y+4, where y is an integer
(2) x=2z+2, where z is an integer

(1): [4(y+1)]^3 - 3 [4(y+1)]^2 + 2 [4(y+1)] = 4^3(y+1)^3 - 3 (4^2) (y+1)^2 + 2 (4) (y+1) --> divisible by 4: sufficient
(2): 2^3(z+1)^3 - 3 (2^2) (z+1)^2 + 2 (2) (z+1) --> divisible by 4: sufficient
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11 Jan 2014, 17:37
Bunuel wrote:
kashishh wrote:
if x is a +ve integer, is x³ - 3x² + 2x divisible by 4?
(1) x = 4y + 4, where y is an integer
(2) x = 2z + 2, where z is an integer

in the above (1) is exactly a multiple for 4, so sufficient BUT in (2) if z = 0 , then x = 2, which when substituted in quest will not be divisible by 4. right?
but the answer is D, both alone sufieicient

can anyone guide. do i have a wrong approach towards DS??

If x is a positive integer , is x^3 - 3x^2+2x divisible by 4?

(1) x=4y+4, where y is an integer --> since x itself is divisible by 4 then x^3-3x^2+2x is divisible by 4. Sufficient.

(2) x=2z+2, where z is an integer --> x^3-3x^2+2x=x(x^2-3x+2)=(2z+2)(4z^2+8z+4-6z-6+2)=4(z+1)(2z^2+z) --> hence this expression is divisible by 4. Sufficient.

As for your question: if x=2 then x^3-3x^2+2x=0. Now, zero is divisible by EVERY integer except zero itself, as 0/integer=integer.

For more on this topic check Number Theory chapter of Math Book: math-number-theory-88376.html

Hope it's clear.

For the second statement could one only say that since x is a multiple of 2 and thus even then x-2 will also be even and that means that E*E = E and multiple of 4?

Thanks

Cheers
J
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Re: If x is a positive integer, is x^3-3x^2+2x divisible by 4? [#permalink]

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11 Jan 2014, 19:17
kashishh wrote:
If x is a positive integer, is x^3-3x^2+2x divisible by 4?

(1) x=4y+4, where y is an integer
(2) x=2z+2, where z is an integer

[Reveal] Spoiler:
in the above (1) is exactly a multiple for 4, so sufficient BUT in (2) if z = 0 , then x = 2, which when substituted in quest will not be divisible by 4. right?
but the answer is D, both alone sufieicient

can anyone guide. do i have a wrong approach towards DS??

Bunuel - could we have solved/approached the question this way?

Given the question stem - x^3-3x^2+2x - factor out an "x" and apply the FOIL method --> therefore we are left with
x(x^2 - 3x +2x) = 0 --> x(x - 2)(x - 1)
Therefore, x must equal 0, 1, or 2

With that said, plug in the values of 0, 1, and 2 into each of statement one and two to determine if they are sufficient
Statement 1 --> only 0 works
Statement 2 --> both 0 and 2 work (due to the value being an integer)

Therefore, OA is D because we can determine what exact values of of the question stems can be valid.

Let me know what you think, just trying to help out with different ways to approach this question.
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Re: If x is a positive integer, is x^3-3x^2+2x divisible by 4? [#permalink]

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12 Jan 2014, 06:52
bparrish89 wrote:
kashishh wrote:
If x is a positive integer, is x^3-3x^2+2x divisible by 4?

(1) x=4y+4, where y is an integer
(2) x=2z+2, where z is an integer

[Reveal] Spoiler:
in the above (1) is exactly a multiple for 4, so sufficient BUT in (2) if z = 0 , then x = 2, which when substituted in quest will not be divisible by 4. right?
but the answer is D, both alone sufieicient

can anyone guide. do i have a wrong approach towards DS??

Bunuel - could we have solved/approached the question this way?

Given the question stem - x^3-3x^2+2x - factor out an "x" and apply the FOIL method --> therefore we are left with
x(x^2 - 3x +2x) = 0 --> x(x - 2)(x - 1)
Therefore, x must equal 0, 1, or 2

With that said, plug in the values of 0, 1, and 2 into each of statement one and two to determine if they are sufficient
Statement 1 --> only 0 works
Statement 2 --> both 0 and 2 work (due to the value being an integer)

Therefore, OA is D because we can determine what exact values of of the question stems can be valid.

Let me know what you think, just trying to help out with different ways to approach this question.

Your approach is not correct. Notice that we are NOT told that x^3-3x^2+2x is 0, thus your derivation that x is 0, 1, or 2 is not right.
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29 Jan 2014, 15:47
Bunuel wrote:
kashishh wrote:
if x is a +ve integer, is x³ - 3x² + 2x divisible by 4?
(1) x = 4y + 4, where y is an integer
(2) x = 2z + 2, where z is an integer

in the above (1) is exactly a multiple for 4, so sufficient BUT in (2) if z = 0 , then x = 2, which when substituted in quest will not be divisible by 4. right?
but the answer is D, both alone sufieicient

can anyone guide. do i have a wrong approach towards DS??

If x is a positive integer , is x^3 - 3x^2+2x divisible by 4?

(1) x=4y+4, where y is an integer --> since x itself is divisible by 4 then x^3-3x^2+2x is divisible by 4. Sufficient.

(2) x=2z+2, where z is an integer --> x^3-3x^2+2x=x(x^2-3x+2)=(2z+2)(4z^2+8z+4-6z-6+2)=4(z+1)(2z^2+z) --> hence this expression is divisible by 4. Sufficient.

As for your question: if x=2 then x^3-3x^2+2x=0. Now, zero is divisible by EVERY integer except zero itself, as 0/integer=integer.

For more on this topic check Number Theory chapter of Math Book: math-number-theory-88376.html

Hope it's clear.

Is it enough to note that we have (x)(x-2)(x-1)

So then if x is even so will (x-2), so basically just knowing that x is even for each statement separately?

Or do we need to go through all the replacing and factorization for each statement?

Thanks!
Cheers!
J
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29 Jan 2014, 22:52
jlgdr wrote:
Bunuel wrote:
kashishh wrote:
if x is a +ve integer, is x³ - 3x² + 2x divisible by 4?
(1) x = 4y + 4, where y is an integer
(2) x = 2z + 2, where z is an integer

in the above (1) is exactly a multiple for 4, so sufficient BUT in (2) if z = 0 , then x = 2, which when substituted in quest will not be divisible by 4. right?
but the answer is D, both alone sufieicient

can anyone guide. do i have a wrong approach towards DS??

If x is a positive integer , is x^3 - 3x^2+2x divisible by 4?

(1) x=4y+4, where y is an integer --> since x itself is divisible by 4 then x^3-3x^2+2x is divisible by 4. Sufficient.

(2) x=2z+2, where z is an integer --> x^3-3x^2+2x=x(x^2-3x+2)=(2z+2)(4z^2+8z+4-6z-6+2)=4(z+1)(2z^2+z) --> hence this expression is divisible by 4. Sufficient.

As for your question: if x=2 then x^3-3x^2+2x=0. Now, zero is divisible by EVERY integer except zero itself, as 0/integer=integer.

For more on this topic check Number Theory chapter of Math Book: math-number-theory-88376.html

Hope it's clear.

Is it enough to note that we have (x)(x-2)(x-1)

So then if x is even so will (x-2), so basically just knowing that x is even for each statement separately?

Or do we need to go through all the replacing and factorization for each statement?

Thanks!
Cheers!
J

$$x^3 - 3x^2+2x=x (x-1) (x-2)$$ --> if x is even, then x-2 is also even, thus x(x-1)(x-2) is divisible by 4.
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Re: If x is a positive integer, is x^3-3x^2+2x divisible by 4? [#permalink]

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31 Mar 2015, 07:16
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Re: If x is a positive integer, is x^3-3x^2+2x divisible by 4? [#permalink]

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14 Nov 2015, 11:48
jlgdr wrote:

Is it enough to note that we have (x)(x-2)(x-1)

So then if x is even so will (x-2), so basically just knowing that x is even for each statement separately?

Or do we need to go through all the replacing and factorization for each statement?

Thanks!
Cheers!
J

This is how I did it,

I think that if we have 3 consecutive integers in the format x(x-1)(x-2) and x is even, then it will be divisible by 4 (2)(3)(4)

If x = Odd it may not be divisible by 4 (3)(2)(1)
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Re: If x is a positive integer, is x^3-3x^2+2x divisible by 4? [#permalink]

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17 Nov 2015, 10:32
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If x is a positive integer, is x^3-3x^2+2x divisible by 4?

(1) x=4y+4, where y is an integer
(2) x=2z+2, where z is an integer

If we modify the question, x^3-3x^2+2x=4t? (t is an integer), --> x(x^2-3x+2)=4t? and we ultimately want to know whether
x(x-1)(x-2)=4t, which is the same as whether x-2 is even
For condition 1, from x=4y+4, x-2=4y+2=2(2y+1). This is even and the condition is sufficient
For condition 2, from x=2z+2, x-2=2z. This is also even and the condition sufficiently answers the question 'yes'

Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.
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Re: If x is a positive integer, is x^3-3x^2+2x divisible by 4? [#permalink]

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24 Aug 2016, 13:30
x^3 - 3x^2+2x divisible by 4
x(X^2-3x+2)
x(x-1)(x-2)

so it is consecutive number.
any three positive consecutive numbers are divisible by 4..
Am i thinking something wrong here?
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Re: If x is a positive integer, is x^3-3x^2+2x divisible by 4? [#permalink]

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26 Aug 2016, 05:43
I simplified the prompt to is x(x-2)(x-1)/4.?

In S1: Any value of y would satisfy the equation, hence it is sufficient
In S2: All the values of z above integer 0 would satisfy the condition, and if I consider z=0, then the numerator becomes 0, and 0 divided by any integer will yield 0. Hence it should be sufficient.

I always pick numbers for testing cases in DS problems. Is that a good practice.?
Re: If x is a positive integer, is x^3-3x^2+2x divisible by 4?   [#permalink] 26 Aug 2016, 05:43
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