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divyalr · Tue Oct 13, 2009 8:53 pm

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

Please add your inputs

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jgreenle · Wed Oct 14, 2009 5:33 pm

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)

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vaibhav.iit2002 · Sun Oct 18, 2009 4:46 am

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)

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