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If a and b are different nonzero integers, what is the value of b? (1 [#permalink]
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04 May 2015, 04:48
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Re: If a and b are different nonzero integers, what is the value of b? (1 [#permalink]
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04 May 2015, 08:18
St 1 says
a^b = ab
1st case a=2 and b=1, works fine 2^1 = 2*1 => 2=2
what if 2^2 = 2*2 => 4=4,
not sufficient.
st#2 says
a^b−a^b−1=2
Again picking numbers a=3 and b=1 => 3  3^0 => 31 =2, works fine.
but if a=2 and b=2 => 2^2  2^(21) => 4  2 =2 , works fine.
not sufficient.
combining both statements same result got two values of b 1 &2.
Answer E



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Re: If a and b are different nonzero integers, what is the value of b? (1 [#permalink]
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04 May 2015, 08:43
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According to statement 1: a^b = ab a^(b1) = b So (a,b) = (2,2) or (any integer, 1) But since a and b are different, the first option can be discarded. So b can only be 1. SUFFICIENT
According to statement 2: a^b  a^(b1) = 2 (a^(b1))*(a1) = 2 Either a^(b1) = 2 and a1 = 1, i.e. (a,b) = (2,2) or a^(b1) = 1 and a1 = 2 i.e. (a,b) = (3,1) or a^(b1) = 2 and a1 = 1 which is not possible as a is nonzero or a^(b1) = 1 and a  1 = 2 i.e. (a,b) = (1,2)
The first option is not possible as a and b are different but second and fourth are possible, which gives us different values of b. NOT SUFFICIENT
A is the right choice here.
Please note that if you remove the condition that a and b are 'different,' then two values of b are possible through statement (1) as well and E becomes the right choice.
Last edited by PrepTap on 05 May 2015, 09:27, edited 1 time in total.



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Re: If a and b are different nonzero integers, what is the value of b? (1 [#permalink]
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05 May 2015, 01:03
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Got tricked initially (did the same mistake), good question !! (need to take note that a,b are different nonzero integers) Q : If a and b are different and nonzero integers , what is the value of b? make sure to use all the information provided in question stem, question stem tells us that
(i) \(a\neq{0}\) or \(b\neq{0}\) (ii) \(a\neq{b}\) now to the statements stmt 1: \(a^{b}\) = ab(a,b) = (1,1), (2,2) , (3,1) , (4,1)
1^1 = 1*1 > ignore this a,b are different nonzero
2^2 = 2*2 > ignore this a,b are different nonzero 3^1 = 3*1 4^1 = 4*1... b is always 1 > b =1 got one solution , so Sufficientstmt 2: \(a^{b}  a^{b1}\)= 2 Simplify : \(a^{b}\) (1\(\frac{1}{a}\)) =2
this equation works for values as above (a,b) = (2,2) , (3,1) (refer below with inserted values)
\(2^{2}\) (1\(\frac{1}{2}\)) => 4 (\(\frac{1}{2}\)) = 2
(2,2) this works But, ignore this a,b are different nonzero
\(3^{1}\) (1\(\frac{1}{3}\)) => 3 (\(\frac{2}{3}\)) = 2
(a,b) = (3,1) works
[Edited] got one more solution
\((1)^{2}\) (1\(\frac{1}{(1)}\)) => 1 (1+1) = 2 (a,b) = (1,2) did works > b got two solution , Not Sufficient
so both stmt 1 & stmt2 are Sufficient: Ans : A
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Last edited by UJs on 07 May 2015, 00:26, edited 6 times in total.



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Re: If a and b are different nonzero integers, what is the value of b? (1 [#permalink]
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05 May 2015, 05:41
UjjwalS wrote: required value of b ? Got tricked initially (did the same mistake ), good question ( need to take note that a,b are different nonzero integers) stmt 1: a^b = ab (a,b) = (1,1), (2,2) , (3,1) , (4,1)
1^1 = 1*1 > ignore this a,b are different nonzero
2^2 = 2*2 > ignore this a,b are different nonzero 3^1 = 3*1 4^1 = 4*1... b is always 1 > b =1 got one solution , so Sufficientstmt 2: a^b  a^b1= 2 Simplify : a^b (1\(\frac{1}{a}\)) =2
this equation works for values as above (a,b) = (2,2) , (3,1) (refer below with inserted values)
2^2 (1\(\frac{1}{2}\)) => 4 (\(\frac{1}{2}\)) = 2
(2,2) this works But, ignore this a,b are different nonzero
3^1 (1\(\frac{1}{3}\)) => 3 (\(\frac{2}{3}\)) = 2
(a,b) = (3,1) works > b got one solution , so Sufficient
so both stmt 1 & stmt2 are Sufficient: Ans : DWhat about a=2 and b=2 2^2 = 4



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Re: If a and b are different nonzero integers, what is the value of b? (1 [#permalink]
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05 May 2015, 08:29
Quote: What about a=2 and b=2
2^2 = 4 viksingh15it is tricky, make sure to use all the information provided in question stem.I understand the confusion. Q : If a and b are different and nonzero integers , what is the value of b? it tells us that 1> \(a\neq{0}\) or \(b\neq{0}\)
2> \(a\neq{b}\)
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Re: If a and b are different nonzero integers, what is the value of b? (1 [#permalink]
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11 May 2015, 05:14
Bunuel wrote: If a and b are different nonzero integers, what is the value of b?
(1) \(a^b = ab\)
(2) \(a^b  a^{b  1} = 2\)
Kudos for a correct solution. MANHATTAN GMAT OFFICIAL SOLUTION:(1) SUFFICIENT: If b=1, then a can be any number whatsoever, because a^1 and a(1) will always be the same number. Therefore the issue is to determine whether b can be any number other than 1. The given equation can be rephrased. Divide both sides by a, giving a^{b1}=b. First, consider the case in which b is positive. If b=2, then this equation reduces to a^1=2, so that a=2; however, this case can be disregarded, because a and b are not allowed to be the same. If b=3 or more, the equation is a^2=3, a^3=4, a^4=5, etc. None of these equations have integer solutions. What if b is negative? If b=1, then a^{2}=1, which is impossible! Something raised to an even power cannot be negative. If b=2 or less, then we have a^{3}=2, a^{4}=3, etc. These equations are all impossible. If the power is even, then the expression can’t be negative. If the power is odd, the resulting value of a will be a fraction. This rules out every case except b=1. The statement is sufficient. (2) NOT SUFFICIENT: To find cases that simplify this equation, it is helpful to factor it. Pull an a^{b1} out of both terms on the right side: a^{b1}(a1)=2. (If you’re not sure why what’s the right factorization, write out the steps yourself; that’ll be a great test of your knowledge of exponent rules). Now test some cases: If a=3 and b=1, then the expression is 3^0(31)=1(2)=2. If a=1 and b=2, then the expression is 1^{3}(11)=(1)(2)=2. There are at least two possibilities for b, so this statement can’t be sufficient. It turns out that if a=1 and b= any even number, you’ll get 2 as the answer (not sure why? test some even numbers to see whether you can figure out the rule). Therefore, b can be 1 or any even number. The correct answer is A.
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Re: If a and b are different nonzero integers, what is the value of b? (1 [#permalink]
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11 May 2015, 06:49
Bunuel wrote: If a and b are different nonzero integers, what is the value of b?
(1) \(a^b = ab\)
(2) \(a^b  a^{b  1} = 2\)
Kudos for a correct solution. a≠b and a ≠0 1) ab=ab a(a^(b1) 1 ) = 0 either a =0 or ab1 =1 if and b are intergers only way possible for this is for b = 1 2) a^b−a^(b−1)=2 a^b ( 1 – 1/a) = 2 a^( b1) * ( a 1) = 2 ( a = 3 and b = 1 )and (a =2 and b = 2) both will satisfy this equation. Answer A



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Re: If a and b are different nonzero integers, what is the value of b? (1 [#permalink]
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