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If x is a positive integer

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If x is a positive integer  [#permalink]

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New post Updated on: 13 Aug 2018, 01:51
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If \(x\) is a positive integer, is the GCD of \(x+3\) and \(x+5\) more than \(1\)?
1. \(3x\) is the common factor of \(12\) and \(6\).
2. \(2x^n\) has \(1\) prime factor.
    A) Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
    B) Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
    C) Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
    D) EACH statement ALONE is sufficient.
    E) Statement (1) and (2) TOGETHER are NOT sufficient.

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Originally posted by EgmatQuantExpert on 28 Feb 2018, 02:08.
Last edited by EgmatQuantExpert on 13 Aug 2018, 01:51, edited 2 times in total.
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Re: If x is a positive integer  [#permalink]

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New post 28 Feb 2018, 03:01
EgmatQuantExpert wrote:

Question:



If \(x\) is a positive integer, is the GCD of \(x+3\) and \(x+5\) more than \(1\)?
1. \(3x\) is the common factor of \(12\) and \(6\).
2. \(2x^n\) has \(1\) prime factor.
    A) Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
    B) Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
    C) Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
    D) EACH statement ALONE is sufficient.
    E) Statement (1) and (2) TOGETHER are NOT sufficient.


we need to find value of x and check if x+3 and x+5 are co-prime together

1) 3x is the common factor of 12 and 6

factors of 12= 1,2,3,4,6,12
factors of 6 = 1,2,3,6

common factor = 1,2,3,6

3x = 3 or 3x=6

x= 1 or x =2

if x=1

x+3 and x+5 are 4 and 6 hcf >1

but if x=2

x+3 and x+5 are 5 and 7 hcf not greater than insufficient


2) 2x^n has 1 prime factor

i.e x^n is a power of 2/x=1

if x=1

2x^n = 2

x=2 n=2

2x^n = 2^3

x=2

similar cases to A

insufficient


combining we get x as 1/2

(E) imo


If common factor = HCF then (A) imo
My understanding of common factor is neccesatrily not the greatest common factor

niks18, pushpitkc and amanvermagmat what do you guys think?
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Re: If x is a positive integer  [#permalink]

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New post 28 Feb 2018, 05:14
Hey Hatakekakashi

A common factor is not necessarily the HCF.
HCF is the highest common factor and there can be other common factors for two numbers as well.
Consider the numbers 4 and 12. The HCF is 4, but 2 is also a common factor.

As per my understanding, the solution you have given is perfect and the OA must be E!
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Re: If x is a positive integer  [#permalink]

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New post 28 Feb 2018, 11:19

Solution:



We need to find:
If the GCD of \(x+3\) and \(x+5\) is more than \(1\) or not.
Statement 1
“\(3x\) is the common factor of \(12\) and \(6\)”.
Factors of \(12= 1, 2, 3, 4, 6,12\)
Factors of \(6= 1,2,3,6\)
Common factor of \(12\) and \(6\) which are in the form \(3x\) are, \(3\) and \(6\).
When,
    \(3x=3\)
      \(x=1\)
    \(3x=6\)
      \(x=2\)
Thus, we do not have a single value of x.
Therefore, Statement 1 alone is NOT sufficient to answer the question.

Statement 2
“\(2x^n\) has \(1\) prime factor “
We know, \(x^n\) has the same number of prime factors as \(x\) has. Therefore,
\(2x\) also has \(1\) factor.
\(2x= 2*x\)
For \(2x\) to have only \(1\) prime factor, the value of \(x\) can be \(1\) or \(2\).
Thus, we do not have a single value of x.
Therefore, Statement 2 alone is NOT sufficient to answer the question.
We are getting the same value of \(x\) from both the statements. Thus, both statements combined will not give the answer.
Therefore, statement 1 and 2 TOGETHER are not sufficient.

Answer: Option E
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Re: If x is a positive integer &nbs [#permalink] 28 Feb 2018, 11:19
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