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Intern  Joined: 20 Oct 2010
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If x is an integer, is (x^2+1)(x+5) an even number?  [#permalink]

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If x is an integer, is (x^2 + 1)(x + 5) an even number?

(1) x is an odd number

(2) Each prime factor of x^2 is greater than 7

Originally posted by mybudgie on 04 Nov 2010, 18:37.
Last edited by Bunuel on 30 Jan 2016, 02:36, edited 2 times in total.
Edited the question
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If x is an integer, is (x^2+1)(x+5) an even number?

The product of 2 integers to be even at least one of them must be even. So if $$x$$ is an odd number then $$(x^2+1)(x+5)=even*even=even$$.

(1) x is an odd number. Sufficient.

(2) Each prime factor of x^2 is greater than 7 --> first of all, prime factors of $$x^2$$ are the same as the prime factors of $$x$$: exponentiation does not "produce" new prime factors. So, each prime of $$x$$ is greater than 7, which means that 2 is not a prime factor of $$x$$, thus $$x$$ is an odd number. Sufficient.

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Re: If x is an integer, is (x^2+1)(x+5) an even number?  [#permalink]

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I could not understand the reason for the explanation of teh second statement...
Can you plsss please explain in a elaborative way...
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Re: If x is an integer, is (x^2+1)(x+5) an even number?  [#permalink]

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jullysabat wrote:
I could not understand the reason for the explanation of teh second statement...
Can you plsss please explain in a elaborative way...

Exponentiation does not "produce" new prime factors means that the prime factors of a positive integer x and the prime factors of x^n (where n is a positive integer) are the same. For example if x=12=2^2*3 then it has two prime factors 2 and 3 and x^2=144=2^4*3^2 also has the same two prime factors 2 and 3.

Now, (2) says: each prime factor of x^2 is greater than 7 --> the prime factors of $$x^2$$, or which is the same the prime factors of $$x$$ can be 11, 13, 17, ... So 2<7 is not a prime factor of $$x$$ which means that $$x$$ must be odd, and as we concluded if $$x$$ is an odd number then $$(x^2+1)(x+5)=even*even=even$$. Sufficient.

Hope it's clear.
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Re: If x is an integer, is (x^2+1)(x+5) an even number?  [#permalink]

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agdimple333 wrote:
if x is an integer, is (x^2 + 1) (x + 5) an even number?

(1) x is an odd number
(2) each prime factor of x ^ 2 is greater than 7

Statement 1 - if x is odd then x+5 is even and hence the expression is even, sufficient
Statement 2 - if each prime factor of x^2 is greater than 7, then all factors are odd and hence x^2 is odd and hence x^2+1 is even making the expression even, sufficient

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GMAT 1: 780 Q51 V48 GRE 1: Q800 V740 Re: If x is an integer, is (x^2+1)(x+5) an even number?  [#permalink]

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The answer should be (D).

We need to state if we have enough information to conclude whether (x^2 +1) (x+5) is an even number.

From statement (1), if x is odd, then x+5 will be even. Therefore (x^2 +1) (x+5) will be (odd +1)*(odd+5) = even * even = even

From statement (2), we know that each prime factor of x^2 is greater than 7. This rules out 2 as being a factor of x^2. Since 2 is the only prime that is also even, this means that x^2 can therefore be expressed entirely as a product of primes, all of which are odd. Therefore x^2 is odd. Therefore (x^2 + 1) is even. Therefore (x^2 + 1) (x+5) is even.

So (D).
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Re: If x is an integer, is (x^2+1)(x+5) an even number?  [#permalink]

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siddhans wrote:
If x is an integer, is (x^2 +1) (x+5) an even number?
(1) x is an odd number
(2) Each prime factor of x^2 is greater than 7

Is $$x$$ even?
OR
Is $$x$$ odd?

(1) $$x$$ is odd. Sufficient.
(2) All prime factors of $$x^2$$ is greater than 7.
Rule: If a number doesn't have at least one 2 as its prime factor, the number will be odd.
Statement 2 is a convoluted way to say that there is NO 2 as a prime factor of $$x$$
OR
$$x$$ is odd. Same as statement 1.
Sufficient.

Ans: "D"
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Re: If x is an integer, is (x^2+1)(x+5) an even number?  [#permalink]

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If x is an integer, is (x^2+1)(x+5) an even number?

(1) x is an odd number

(2) Each prime factor of x^2 is greater than 7

Originally posted by pbull78 on 28 Jan 2012, 05:56.
Last edited by Bunuel on 23 May 2013, 04:53, edited 2 times in total.
Edited the question.
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Re: If x is an integer, is (x^2+1)(x+5) an even number?  [#permalink]

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pbull78 wrote:
If x is an integer, is (x2+1)(x+5) an even number?
1). x is an odd number. 2). each prime factor of x2 is greater than 7

although discussed but still need explanation of this not able to understand .......... 2) each prime factor of x2 is greater than 7

pbull78 you please format the question properly. Question should read:

If x is an integer, is (x^2+1)(x+5) an even number?

(1) x is an odd number --> (x^2+1)(x+5)=(odd+odd)(odd+odd)=even*even=even. Sufficient.

(2) Each prime factor of x^2 is greater than 7 --> 2 is not a prime factor of x^2, so not a prime factor of x as well --> x=odd --> the same as above. Sufficient.

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Re: If x is an integer, is (x^2+1)(x+5) an even number?  [#permalink]

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i copied this question from source as it is

but still i am not able to understand this point

2) Each prime factor of x^2 is greater than 7 --> 2 is not a prime factor of x^2, so not a prime factor of x as well --> x=odd --> the same as above.
Can u expalin with example ?
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Re: If x is an integer, is (x^2+1)(x+5) an even number?  [#permalink]

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pbull78 wrote:
i copied this question from source as it is

but still i am not able to understand this point

2) Each prime factor of x^2 is greater than 7 --> 2 is not a prime factor of x^2, so not a prime factor of x as well --> x=odd --> the same as above.
Can u expalin with example ?

Each prime factor of x^2 is greater than 7 --> as 2 is less than 7, then 2 is not a prime factor of x^2, which means that 2 is not a prime of x as well, because if it is a prime of x then x^2 would also have it.

Or:
Each prime factor of x^2 is greater than 7 --> as 2 is less than 7, then 2 is not a prime factor of x^2, which means that x^2 is an odd number --> x^2=odd --> x^2=x*x=odd --> x=odd.

Or:
Each prime factor of x^2 is greater than 7 --> primes more than 7 are odd (in fact the only even prime is 2) --> x is a product of some odd primes more than 7 --> x^2 is an odd number --> x=odd.

Hope it's clear.
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Re: If x is an integer, is (x^2+1)(x+5) an even number?  [#permalink]

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If x is an odd number, x^2 must be odd => (x^2+1) must be even.
By the same logic, (x+5) is even. As addition of two odd numbers results in an even number.
Multiplication of two even numbers results in an even number. Thus (x^2+1)(x+5) is an even number.

Satetment (1) is SUFFICIENT.

And for (2), each prime factor of x^2 is greater than 7 implies that 2 is not a prime factor of x^2 and in turn 2 is not a prime factor of x. (As all the prime factors of x^2 are also prime factors of x)

That means x is composed of 11, 13, 17, 19, 23 etc (Prime numbers greater than 7).
Therefore x is an odd number => Equivalent to statement (1).

Statement (2) is SUFFICIENT.

The correct answer is D.
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Re: If x is an integer, is (x^2+1)(x+5) an even number?  [#permalink]

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In statement 2. Can X not be 0?
each prime number of X^2 is greater than 7. Understanding that 2 is the only prime number so if the prime factor of x^2 is above 7 then the only numbers we are left with are odd numbers. Since 0 a multiple of every number. Can x be 0?

Maybe it's too dumb of a question but hey we are all here to learn. Thanks
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Re: If x is an integer, is (x^2+1)(x+5) an even number?  [#permalink]

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Rocket7 wrote:
In statement 2. Can X not be 0?
each prime number of X^2 is greater than 7. Understanding that 2 is the only prime number so if the prime factor of x^2 is above 7 then the only numbers we are left with are odd numbers. Since 0 a multiple of every number. Can x be 0?

Maybe it's too dumb of a question but hey we are all here to learn. Thanks

x cannot be 0 because 0 is divisible be primes which are less than 7 (2, 3, and 5).
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