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If a, b and x are integers such

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If a, b and x are integers such [#permalink]

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Q.

If a, b and x are integers such that \(a^6=b^3=\frac{|x|}{x}\), what is the value of a - b

(1) \(a^3b^7 > 0\)

(2) \(a + b > 0\)


Answer Choices



A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.


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Originally posted by EgmatQuantExpert on 24 May 2017, 11:45.
Last edited by EgmatQuantExpert on 31 May 2017, 05:26, edited 1 time in total.
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If a, b and x are integers such [#permalink]

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New post Updated on: 31 May 2017, 05:43
The official solution has been posted. Looking forward to a healthy discussion..:)
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Originally posted by EgmatQuantExpert on 24 May 2017, 11:47.
Last edited by EgmatQuantExpert on 31 May 2017, 05:43, edited 1 time in total.
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Re: If a, b and x are integers such [#permalink]

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New post 24 May 2017, 12:00
1
IMO D
\(a^6 = b^3 = \frac{|x|}{x}\)

\(\frac{|x|}{x}\) = 1 or -1 depending on the sign of x,but since this fraction is equal to \(a^6\),it has to be equal to 1.

Therefore x=some postive,b =1 and a =+1 or -1 => a-b=0 or a-b=-2 depending on the value of a.

statement 1:

\(a^3b^7\)>0,

since b=1, \(a^3\)>0, not possible for this statment.

therefore a>0 => a=1.

a-b=0


statement 2:

a+b >0 ,

if a=-1,then a+b=0. Therefore a=1.

a-b=0.

Both the statement satisfies ,so D.
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Re: If a, b and x are integers such [#permalink]

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New post 31 May 2017, 05:42

Official Solution



Steps 1 & 2: Understand Question and Draw Inferences

Given: Integers a,b,x

    • \(a^6=b^3=\frac{|x|}{x}\)
To Find: Value of a - b ?

\(\frac{|x|}{x}\) can take two possible values:

    • If \(x > o, |x| = x, \frac{|x|}{x}=\frac{x}{x}=1\)

    • If \(x < o, |x| = -x, \frac{|x|}{x}=−\frac{x}{x}=−1\)

    • \(a^6=\frac{|x|}{x}\)

    • As \(a^6\) is always positive,\(a^6 = 1\), i.e. \(a = 1\) or \(-1\)

    • So, we can reject the value of \(\frac{|x|}{x}=−1\)

    • \(b^3=\frac{|x|}{x}=1\)

    • b = 1

Possible values of \(a – b\)

    • If \(a = 1\) and \(b = 1, a – b = 0\)

    • If \(a = -1\) and \(b = 1, a- b = -2\)

So, we need to find the unique value of a to find the value of \(a – b\).

Step 3: Analyze Statement 1 independently

(1) \(a^3*b^7 > 0\)

    • Rewriting \(a^3b^7\) as \(ab(a^2b^6)\)

    • Therefore, \(ab(a^2b^6)>0\)

    • We know that \(a^2b^6\) is always \(> 0\) (even power of any number is always positive)

      • So, for \(ab(a^2b^6)> 0\)

      • \(ab > 0\)

    • This tells us that a and b have same signs.

    • Since \(b > 0\), therefore a will also be greater than 0, so the value of \(a = 1\).

      • \(a – b = 1 -1 = 0\)

Sufficient to answer

Step 4: Analyze Statement 2 independently

(2) \(a + b > 0\)

    • If \(a = 1\) and \(b = 1\), \(a + b = 2 > 0\)

    • If \(a = -1\) and \(b = 1\), \(a + b = 0\), is not greater than zero

      • Hence, we have a unique answer, where \(a =1\) and \(b = 1\)

    • Thus \(a – b = 1 – 1 = 0\).

Sufficient to answer.

Answer: D


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Re: If a, b and x are integers such [#permalink]

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New post 24 Aug 2017, 21:43
Since \(a^6\) >=0
\(\frac{|x|}{x}\) will be positive, and |x| >=0 so x must be positive
=> |x| = x (also x can't be 0) and the fraction can be reduced to 1.

\(b^3\)=1 => b = 1
\(a^6\) = 1 => a =1, or -1

After analyzing, we can see the true question here is whether a is 1 or -1.

1- \(a^3\) \(b^7\) >0
or \(a^3\) x 1 >0
so a>0
=> a=1
SUFF

2-a+b>0
or a+1 > 0
a can only be -1 or 1, so a =1
SUFF

In sum, answer is D.
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If a, b and x are integers such [#permalink]

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New post 26 Aug 2017, 11:24
From the question, without looking at the statements, we can infer that the a=b=1 due to |x|/x. It cannot be 0 as 0/0 is undefined. It can't be -1 too due to b^3. Is this question legit?


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Re: If a, b and x are integers such [#permalink]

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New post 27 Aug 2017, 02:42
antztheman wrote:
From the question, without looking at the statements, we can infer that the a=b=1 due to |x|/x. It cannot be 0 as 0/0 is undefined. It can't be -1 too due to b^3. Is this question legit?


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b must be 1 but a can be 1 ass well as -1. For both values, 1 and -1, a^6 is still 1.

Possible values of \(a – b\)

    • If \(a = 1\) and \(b = 1, a – b = 0\)

    • If \(a = -1\) and \(b = 1, a- b = -2\)

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Re: If a, b and x are integers such [#permalink]

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New post 27 Aug 2017, 15:57
I thought a^6 = |x|/x => as a6 is positive x has to be positive: => a = 1?
x>0 => be is +1 and x<0 means b= -1

Is my reasoning wrong?

Can someone help me..
Thanks,
Re: If a, b and x are integers such   [#permalink] 27 Aug 2017, 15:57
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