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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] New post 06 Feb 2012, 12:30
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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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Re: data sufficiency [#permalink] New post 06 Feb 2012, 12:44
Expert's post
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.

Answer: D.

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] New post 20 Feb 2013, 20:38
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] New post 21 Feb 2013, 02:30
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Re: If x is a positive integer, is x^3-3x^2+2x divisible by 4? [#permalink] New post 26 Feb 2013, 19:16
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
==> Answer is D
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Re: data sufficiency [#permalink] New post 11 Jan 2014, 16:37
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.

Answer: D.

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

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Re: If x is a positive integer, is x^3-3x^2+2x divisible by 4? [#permalink] New post 11 Jan 2014, 18:17
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] New post 12 Jan 2014, 05:52
Expert's post
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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PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: data sufficiency [#permalink] New post 29 Jan 2014, 14:47
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.

Answer: D.

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?

Please advice
Thanks!
Cheers!
J :)
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Re: data sufficiency [#permalink] New post 29 Jan 2014, 21:52
Expert's post
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.

Answer: D.

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?

Please advice
Thanks!
Cheers!
J :)


Didn't you answer your own question?

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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NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: data sufficiency   [#permalink] 29 Jan 2014, 21:52
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