If a and b are two distinct positive integers, what is the largest num

Question

If a and b are two distinct positive integers, what is the largest number that leaves no remainder when it divides a and b?

(1) a and b can be written as 8x and 8y respectively, where x and y are distinct integers with no common prime factors.

(2) a is equal to 24 and both a and b are divisible by the same perfect cubes.

AvidDreamer09 · Answer

Question stem clue 1: a≠b 
Clue 2: What is largest number that leaves no remainder is another way of asking the greatest common factor (gcf) of a and b

(1) a and b can be written as 8x and 8y respectively, where x and y are distinct integers with no common prime factors.

a=8x and b=8y

x and y are distinct with no common prime factors, so the largest multiple common to both a&b has to be 8 as it does have have any other common prime factors. Sufficient

(2) a=24 and both a&b are divisible by the same perfect cubes.

Only restriction is that a≠b 
a=24=8*3= (2^3)*3 (only possible perfect cube is 8)

b can be a multiple of 8 that is greater than 24. 
a=8*3 and if b=8*8*3 GCF of a and b is 24
a=8*3 and if b=8*5 GCF of a and b 8
Not sufficient

Hence A

TarPhi · Answer

Statement 1:
a=8x and b=8y

x and y are distinct with no common prime factors, so the largest multiple common to both a&b has to be 8 as it does not have any other common prime factors. Sufficient

Statement 2:
a=24 and both a&b are divisible by the same perfect cubes.

The only restriction is that a≠b
a=24=8*3= (2^3)*3 (only possible perfect cube is 8)

b can be a multiple of 8 that is greater than 24.
a=8*3 and if b=8*8*3 GCF of a and b is 24
a=8*3 and if b=8*5 GCF of a and b 8
Not sufficient

Hence A

CrackverbalGMAT · Answer

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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If a and b are two distinct positive integers, what is the largest num

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