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x is a positive integer greater than two; is (x^3 + 19837)(x^2 + 5)(x – 3) an odd number?

(1) The sum of any prime factor of x and x is even (2) 3x is an even number

Source: Platinum GMAT Kudos for a correct solution.

Question : is (x^3 + 19837)(x^2 + 5)(x – 3) an odd number?

For this result to be odd, every part i.e. (x^3 + 19837), (x^2 + 5) and (x – 3) must be ODD [Because Even + Odd = Odd] Which is possible only whn x is EVEN

Question Rephrased : Is x Even?

Statement 1:The sum of any prime factor of x and x is even

Let, x is odd: Then Every Prime factor of x must be Odd But x + factor = Even i.e. odd + Odd = Even i.e. X can be Odd

Similarly, Let, x is Even: Then One of the Prime factors of x must be e (even) But x + factor = Even i.e. Even + Even = Even i.e. X can be Even as well

NOT SUFFICIENT

Statement 2:3x is an even number

if 3x is even then x must be Even

SUFFICIENT

answer: Option B
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x is a positive integer greater than two; is (x^3 + 19837)(x^2 + 5)(x [#permalink]

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26 Jun 2015, 08:51

2

This post received KUDOS

is (x^3 + 19837)(x^2 + 5)(x – 3) an odd number?

Properties to remember: abc = odd, if all three variables are odd. abc = even, if any of variables is even.

(x^3 + 19837) = if x is odd, the sum will be even, if x is even the sum will be odd. (x^2 + 5) = if x is odd, the sum will be even, if x is even the sum will be odd. (x – 3) = if x is odd, the difference will be even, if x is even the difference will be odd.

So, the question: is the X - even?

(2) 3x is an even number

Start with the simpler looking statement, is 3x is even, then according to simple arithmetics rules - x is even. Sufficient. Interestingly, this statement is so easy, that it looks like a B-trap (Some might think "this is too easy, I bet A is also sufficient and the answer is D")

(1) The sum of any prime factor of x and x is even

This is the tricky one, so what does this say?

We are given that X is greater than 2, so if X is any odd number than some of X and any of its odd primes will be even (ODD + ODD = EVEN). But, if X is even, than we have 2 scenarios: it can have ODD and EVEN prime factors, meaning that it doesn't meet Statement 1 requirements, and this is where the trap lies: don't forget that X can be a power of 2! hence, it can have only one prime number, and it will be EVEN + EVEN = EVEN! So, we can't narrow down this to ODD or EVEN. Insufficient.

I think that the following elements are created to lure students into a trap of choosing D: 1. Statement 2 is very easy as compared to Statement 1 and complex question stem (to imply that there is a B-trap) 2. We are given that x is greater than 2, though it doesn't matter for question (to confuse students into forgetting that powers of 2 are needed to be taken into consideration when solving Statement 1).
_________________

GMAT questions don't come with information just to confuse the test takers. So please understand that if GMAT gives an information then it's either to define the question within the acceptable boundaries or to use for solving the question. It's never to confuse the test takers.
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Re: x is a positive integer greater than two; is (x^3 + 19837)(x^2 + 5)(x [#permalink]

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26 Jun 2015, 09:15

GMATinsight wrote:

GMAT questions don't come with information just to confuse the test takers. So please understand that if GMAT gives an information then it's either to define the question within the acceptable boundaries or to use for solving the question. It's never to confuse the test takers.

Could you please help me with why the information about X being > 2 was given? I might be missing something. I mean, it doesn't confuse you, but you don't need it to solve the question.
_________________

x is a positive integer greater than two; is (x^3 + 19837)(x^2 + 5)(x [#permalink]

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26 Jun 2015, 09:34

I selected option D based on the usage of the word any

Statement 2 is straightforward so there is no need for any elaboration on that.

The sum of any prime factor of x and x is even

I read it like this: If a number has three prime factors, the three sums, i.e sum of each prime factor of x taken one at a time and the number x is even. This can only happen if the number are odd. So this statement is also sufficient. Hence D.

GMAT questions don't come with information just to confuse the test takers. So please understand that if GMAT gives an information then it's either to define the question within the acceptable boundaries or to use for solving the question. It's never to confuse the test takers.

Could you please help me with why the information about X being > 2 was given? I might be missing something. I mean, it doesn't confuse you, but you don't need it to solve the question.

Hi kvazar, I am not saying you are wrong. The comment I mention was a general understanding about GMAT questions so that the message goes right to the readers that "any information of question that you are not able to relate isn't to confuse you, it's to help you understand/solve the question better and without using it your answer has very high potential to be incorrect". Also, this question hasn't come from GMAC so it may have have such flaws but not the questions of GMAC.
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Re: x is a positive integer greater than two; is (x^3 + 19837)(x^2 + 5)(x [#permalink]

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26 Jun 2015, 10:28

akashganga wrote:

I selected option D based on the usage of the word any

Statement 2 is straightforward so there is no need for any elaboration on that.

The sum of any prime factor of x and x is even

I read it like this: If a number has three prime factors, the three sums, i.e sum of each prime factor of x taken one at a time and the number x is even. This can only happen if the number are odd. So this statement is also sufficient. Hence D.

DJ

And this is the trap I was talking about! If X = 8, then it has only one prime factor = 2, and 2 + 8 = 10 (EVEN). It is not necessary for X to have odd prime factors!
_________________

Re: x is a positive integer greater than two; is (x^3 + 19837)(x^2 + 5)(x [#permalink]

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26 Jun 2015, 10:34

GMATinsight wrote:

kvazar wrote:

GMATinsight wrote:

GMAT questions don't come with information just to confuse the test takers. So please understand that if GMAT gives an information then it's either to define the question within the acceptable boundaries or to use for solving the question. It's never to confuse the test takers.

Could you please help me with why the information about X being > 2 was given? I might be missing something. I mean, it doesn't confuse you, but you don't need it to solve the question.

Hi kvazar, I am not saying you are wrong. The comment I mention was a general understanding about GMAT questions so that the message goes right to the readers that "any information of question that you are not able to relate isn't to confuse you, it's to help you understand/solve the question better and without using it your answer has very high potential to be incorrect". Also, this question hasn't come from GMAC so it may have have such flaws but not the questions of GMAC.

I agree! My experience is that if you didn't use ALL the information, then chances are you're missed something. But in my reasoning for this question I just focused on what I have at hand, not what GMAC had at it's mind. But I'm pretty sure are numerous other questions with excess information on the forum. I won't be surprised if they are not GMAC created. Maybe Bunuel can shed some light, if he ever met questions like this created by GMAC.
_________________

x is a positive integer greater than two; is (x^3 + 19837)(x^2 + 5)(x – 3) an odd number?

(1) The sum of any prime factor of x and x is even (2) 3x is an even number

Source: Platinum GMAT Kudos for a correct solution.

Platinum GMAT Official Solution:

In order to solve this question efficiently, it is necessary to begin with number properties. For a product of any number of terms to be odd, all the terms must be odd. If there is but one even term, the product will be even. To see this, consider the following examples: All Terms Odd --> Odd Product 3*7*9*5 = 945 7*9*3 = 189 1*3*5 = 15

But: One or More Even Terms --> Even Product 3*7*9*2 = 378 7*9*4 = 252 1*3*5*6 = 90

In order for (x^3 + 19837)(x^2 + 5)(x – 3) to be an odd number, all the terms must be odd.

To determine under what conditions each term will be odd, it is important to remember the following relationships: odd + odd = even odd - odd = even

even + even = even even - even = even

even + odd = odd even - odd = odd odd + even = odd odd - even = odd

The only way for each term of (x^3 + 19837)(x^2 + 5)(x – 3) to be odd is if an even and an odd number are added or subtracted together within the parenthesis of each term. In other words: even + odd = odd: For (x^3 + 19837) to be odd, since 19837 is odd, x^3 will need to be even. This will happen only when x is even.

even + odd = odd: For (x^2 + 5) to be odd, since 5 is odd, x^2 will need to be even. This will happen only when x is even.

even - odd = odd: For (x – 3) to be odd, since 3 is odd, x will need to be even.

When combining the results from the analysis of the three terms above, the only way for (x^3 + 19837)(x^2 + 5)(x – 3) to be odd is if each term is odd. This will only happen if x is even. Consequently, the original question can be simplified to: is x even? Another version of the simplified question is: what is the parity of x?

Evaluate Statement (1) alone.

In order for the sum of any prime factor of x and x to be even, it must follow one of two patterns: Pattern (1): even + even = even Pattern (2): odd + odd = even

There are two possible cases:

Case (1): x is even. In this case, Pattern (1) must hold. Since x is even in Case (1), any and every prime factor of x must be even (otherwise we could choose an odd prime factor of x and the sum of x and the odd prime factor would be odd). Let's consider some examples: Let x = 12: However, x cannot equal 12 since one prime factor of 12 is 3 and 12 + 3 = odd number. Let x = 26: However, x cannot equal 26 since one prime factor of 26 is 13 and 26 + 13 = odd number. Let x = 14: However, x cannot equal 14 since one prime factor of 14 is 7 and 14 + 7 = odd number. Let x = 16: x can equal 16 since every prime factor of 16 is even and as a result we know that and 16 + any prime factor = even number. It is clear that Statement (1) allows x to be even (e.g., 16 is a possible value of x).

Case (2): x is odd. In this case, Pattern (2) must hold. Since x is odd in Case (2), any and every prime factor of x must be odd (otherwise we could choose an even prime factor of x and the sum of x and the even prime factor would be odd). Since all the prime factors of x are odd, x must be odd in Case (2). Let's consider some examples: Let x = 11: Every prime factor of 11 is odd, so: 11 + prime factor of 11 = even number. Let x = 15: Every prime factor of 15 is odd, so: 15 + prime factor of 15 = even number. Since Statement (1) allows x to be either even (e.g., 16) or odd (e.g., 15), we cannot determine the parity of x.

Statement (1) is NOT SUFFICIENT.

Evaluate Statement (2) alone. 3x = Even Number (odd)(x)=(even) x must be even because, as shown above, if x were odd, 3x would be odd.

Statement (2) is SUFFICIENT since it definitively tells the parity of x.

Since Statement (1) alone is NOT SUFFICIENT but Statement (2) alone is SUFFICIENT, answer B is correct. _________________

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Re: x is a positive integer greater than two; is (x^3 + 19837)(x^2 + 5)(x [#permalink]

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29 Jun 2015, 14:55

X>2, positive integer; Is (X^3+oddnumber)(X^2+Odd)(X-Odd) = odd? If X= even , then expression will be (e+o)(e+o)(e-o) =o*o =Odd If X= odd , then expression will be (o+o)(o+o)(o-o) = there is one even, so entire expression will be Even

1. X can be 4, so prime factors of 4 will 2 +4=Even, then expression will be Odd. But when X=5 sum of prime factor will be 5+5=even, but expression will be Even. So insufficient 2. 3X is even, that means X must be even. So expression will be Odd. Sufficient

x is a positive integer greater than two; is (x^3 + 19837)(x^2 + 5)(x [#permalink]

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29 Nov 2017, 10:58

Simplifying big question: \((x^3 + 19837)(x^2 + 5)(x – 3)\) = odd? \((x^3 + odd)(x^2 + odd)(x - odd)\) = odd? basically question is x even ?

Statement 1: The sum of any prime factor of x and x is even x can be square of odd prime, say 49, prime factor 7 + 49 = 56 (even) => x is odd or x can be any power of 2, say 64, prime factor 2 + 64 = 66 (even) => x is even

x can be odd or even -> Not Sufficient

Statement 2: 3x is even since x is integer, x must be even integer, to make 3x even -> Sufficient.