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Note that we are not told that the variables are integers only.

(1) 5^a = 25 --> \(a=2\), but we can not get the values of \(b\). Not sufficient.

(2) c = 36 --> \(3^a*4^b = c\): it's tempting to write \(3^2*4^1=36\) and say that \(b=1\) but again we are not told that the variables are integers only. So, for example it can be that \(3^a=36\) for some non-integer \(a\) and \(b=0\), making \(4^b\) equal to 1 --> \(3^a*4^b =36*1=36\). Not sufficient.

(1)+(2) As \(a=2\) and \(c = 36\) then \(9*4^b=36\) --> \(b=1\). Sufficient.

for Q1) my reasoning was since 3^a*4^b = c and base 3 is "even" and base 4 is "odd" then there wont be any other answer except a=2 and b=1. But I was wondering what are the other values of a and b can be for this equation to be true. Sorry, I am asking too much

Thanks for the awesome explanation. You rock Bunuel!
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for Q1) my reasoning was since 3^a*4^b = c and base 3 is "even" and base 4 is "odd" then there wont be any other answer except a=2 and b=1. But I was wondering what are the other values of a and b can be for this equation to be true. Sorry, I am asking too much

For 1: equation \(3^a*4^b = 36\) (if there is no restrictions on a and b) has infinitely many solutions for a and b: a=0 --> 4^b=36 --> b=log_4(36); a=1 --> 4^b=12 --> b=log_4(12), ...

Thanks Bunuel. Makes sense, the power of logarithms! I was wondering whether the first question was 700+. This is a new perspective. I havent really used logs in any of questions. Awesome !
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Thanks Bunuel. Makes sense, the power of logarithms! I was wondering whether the first question was 700+. This is a new perspective. I havent really used logs in any of questions. Awesome !

Actually you don't really need to use logarithms (this concept is not tested on GMAT), you should just realize that since you are not told that the variables are integers only then for example if \(3^a=3^0=1\) then there will exist some non-integer \(b\) for which \(4^b =36\) thus you can not get single numerical value of \(b\) from statement (2).
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But how to disprove, 3^2 * 2^2 = 36, even though it's not mentioned that x and y are not integers ? In that case, answer is B ! (1) is not sufficient, as c is not known.
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Formula of Life -> Achievement/Potential = k * Happiness (where k is a constant)

if we can assume then we can also assume a is not =0 then !!!!

Your assumption has no bearing on the inference from Statement 2). The fact remains that 2) is insufficient data and the nature of b is unknown , it may or may not be integer.

If you are wondering why stmnt 2 alone is not sufficient, think of it this way:

\(3^a4^b = c\) (2) c = 36

So \(3^a4^b = 36\) Now for every value of a, there is a different value of b. Say, a = 1, then 4^b = 12 and b = 1.79 approx a = 2, then 4^b = 4 and b = 1 a = 3, then 4^b = 36/27 and b = 0.2 approx and so on...

If we were given that a and b are integers, then answer would have been (B)
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If you are wondering why stmnt 2 alone is not sufficient, think of it this way:

\(3^a4^b = c\) (2) c = 36

So \(3^a4^b = 36\) Now for every value of a, there is a different value of b. Say, a = 1, then 4^b = 12 and b = 1.79 approx a = 2, then 4^b = 4 and b = 1 a = 3, then 4^b = 36/27 and b = 0.2 approx and so on...

If we were given that a and b are integers, then answer would have been (B)

Dear Karishma Agreed with your approach i also think that way

but wanted to ask you one thing that does GMAT wants us to seriously think the nos to be real nos until specifically stated to be integers or this kind of confusion is rare in real GMAT or these kind of questions are actually TRAP questions
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I broke down the numbers to prove that b=1, but I think you can confidently answer C without having to do so.

And to answer your question ^^, yes the GMAT will try to trick you, so be wary of whether or not A or B need to be integers. If they do then it completely changes the answer...always be on the lookout for any restrictions regarding integers.

Dear Karishma Agreed with your approach i also think that way

but wanted to ask you one thing that does GMAT wants us to seriously think the nos to be real nos until specifically stated to be integers or this kind of confusion is rare in real GMAT or these kind of questions are actually TRAP questions

Sorry to say but yes, GMAT wants you to consider whether the number can be non-integer and if so, how your answer could vary. Also, GMAT is not above laying such and many other traps for you e.g. the easy C. This is one of the reasons why DS questions are considered much harder than PS questions. You have to analyze the problem from every aspect and consider every possibility. That is why it takes more time to do DS questions even though the concepts behind PS and DS questions are exactly the same. While practicing these questions, there will be many times when you will feel like pulling your hair out because you overlooked one tiny thing e.g. 'what if x is 0' or something like that and all effort was wasted because you got the answer wrong. Hence, try and be aware of these traps and practice as much as you can.
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