All possible roots lie between 1-10, inclusive, because 10 to the power of 3 is 1,000, which is the max for n. So starting from either 1 or 10 and working through, you find the following:

1 squared is 1 and 1 cubed is also 1, so n can equal 1
2 squared is 4 and 2 cubed is 8, 8 does not have an perfect square, so n cannot equal 8.
3 squared is 9 and 3 cubed is 27, 27 does not have a perfect square so n cannot equal 27.
4 squared is 16 and 4 cubed is 64, and 64 has a square root of 8, so n can equal 64.
5 squared is 25 and 5 cubed is 125, 125 does not have a perfect square, so n cannot equal 125.

and so on, until you find that 10 squared is 100 and 10 cubed is 1000, so n can equal 1000.

the total number of possible n's is 3 (1,64,1000)

to be a perfect square and perfect cube, no. will have to be of form a^6

if a = 1, a^6=1 < 1000
if a = 2, a^6=64 < 1000
if a = 3, a^6=729 < 1000
if a = 4, a^6=4096 > 1000

hence total 3 no.s (1,64,729)

divyalr wrote:A perfect square is defined as the square of an integer and a perfect cube is defined as the cube of an integer. How many positive integers n are there such that n is less than 1,000 and at the same time n is a perfect square and a perfect cube?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

We note that if an integer is a perfect square and a perfect cube at the same time, then it is the sixth power of some integer.

We need to determine how many numbers exist that when raised to the 6th power are less than 1,000.

1^6 = 1 < 1000

2^6 = 64 < 1000

3^6 = 729 < 1000

4^6 = 4,096 > 1000

Since 4^6 > 1000, we see that only 3 values exist.

Answer: B