From the question data, we know that a and b are distinct positive integers. We have to find the HCF of a and b.
From statement I alone, a and b can be written as 8x and 8y respectively; x and y are distinct integers with no common prime factors. This means that x and y are co-prime.
Any two numbers a and b can always be written as,
a = h*f1 and b = h*f2, where h is the HCF of the numbers and f1 & f2 are co-primes. We can now conclude that 8 is the HCF of a and b. Statement I alone is sufficient to find the HCF of a and b. Answer options B, C and E can be eliminated. The possible answer options are A or D.
From statement II alone, a=24 and both a and b are divisible by the same perfect cubes. Now, there can only be 2 such perfect cubes i.e. 1 and 8. So, essentially, we are saying that a and b are either both divisible by 1 or 8. By saying this, we are saying that the HCF can be 1 or 8. Clearly, this is insufficient to find a unique value of the HCF.
For example, if a = 24 and b = 17, HCF = 1. Note that both 24 and 17 are divisible by the same perfect cube 1.
On the other hand, if a = 24 and b = 8, HCF = 8. In this case also, both 24 and 8 are divisible by the same perfect cube 8.
Statement II alone is insufficient, answer option D can be eliminated.
The correct answer option is A.
Hope that helps!
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