An important note here is that 1 is a factor of all integers (because it evenly divides).

since n > 0 and n^2 is an integer, it indicates that n is an integer.

integer/1 = remainder of 0

1 to the power of whatever is 1, so if n is 1 then 1/1 = remainder 0 and 1^(1/2) = 1/1 = remainder 0.

Now if any other power, we will still have integer/1 = remainder of 0, however if the square root is not an integer then there will be remainder. Hence, the answer is D.

To think of it as a matter of understanding the underlying concept rather than a strategy of picking numbers would follow this logic:

integer/1 = remainder of 0
not integer/1 = some remainder

This is because 1 is a factor of all numbers as mentioned above.

I would prefer to use examples here. For example if n = 4, n^2 is an int and:

res 4/1 = 0

res sqrt(4) = 0

but if n = 3

res n/1 = 0

res sqrt(3) = 0.7777

I think that the approach of: If n^2 is an integer then n is an integer is not correct because we could apply the same for the second condition and say that as n is an integer then sqrt(n) is an integer.

You raise a great point; however, the only case in which the condition you are stating is true is if n = 1. Because every other positive integer in a sqrt will not yield an integer.

I hope this makes sense.

@Salsanousi I think jgastelor is right when he says that n^2 integer doesn't imply n is an integer. You could take n^2 equal every possible integer (like 1,2,3,4,5 to start with) and then n will be equal to +square root of these numbers and not necessarily an integer...

This problem doesn't make sense.
We don't consider remainders for the division of real numbers.
Since we are not sure $\sqrt{n}$ is an integer, Quantity B, the remainder when $\sqrt{n}$ is divided by $1$ doesn't make sense.
A remainder is defined for integers only.

Remember: Remainder is always lesser than divisor.

Quantity A: 0

Quantity B: Value between 0 and 1, that is, 0 <= Quantity B < 1.


Thus we are sure that sometimes A will be greater, and sometimes B, and sometimes both will be equal. Hence relationship can not be determined.

Answer D.

Remember: Remainder is always lesser than divisor.

Quantity A: 0

Quantity B: Value between 0 and 1, that is, 0 <= Quantity B < 1.


Thus we are sure that there is sometimes A will be greater, and sometimes B, and sometimes both will be equal. Hence relationship can not be determined.

Answer D.