If a and b are positive integers, is ∛ab an integer?
(1) √a is an integer
(2) b= √a
The question is invariably asking us to determine whether the expression inside the radical, ab, is a perfect cube. One basic property of a perfect cube which will be very useful in problems like these, is that every perfect cube contains a triplet of prime factors. For example,
8 = 2* 2*2; 125 = 5*5*5; 216 = 2*2*2*3*3*3
Hence, in this question,we should be able to find out if the expression ab can give us triplet/s of prime factors.
The first statement, when considered alone, is quite obviously insufficient to answer the question with a definite YES or a definite NO since it does not provide any information about 'b'. The only conclusion we can draw from this statement is that 'a' is a perfect square. So, a can take values of 1 or 4 or 9 and so on. However, 'ab' being a perfect cube depends on the specific value of 'b' which is why the first statement is not sufficient when taken alone. For example,
If a=4 and b=2, ab = 8 is a perfect cube and we can answer the main question with a Yes. However, if a=4 and b=3, ab= 12 which is not a perfect cube and we can answer the main question with a No.
From the second statement, we can gather that 'a' should be a perfect square since the square root of 'a' is giving us 'b', which is also an integer (given in the question statement). Also, we may say that b^2=a since b= √a (squaring both sides). Therefore, ab = b^3 which is definitely a perfect cube since 'b' is an integer. As such, the second statement alone is sufficient to answer the question asked with a definite Yes.
The easiest trap that one can fall into in such questions, is to assume that the two statements have to be combined to find out the answer. This can happen when you have not fully processed the data given in the question statement and hence not retained it during the solution.
Option B is the answer.
Cheers,
Crackverbal Academics Team
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