square and cube

[kushal.adhia]

If x is prime and nx is both the square of an integer and the cube of an integer, where n is a positive integer, what is the greatest possible value of (1/nx)?

A. 1
B. 1/32
C. 1/64
D. 1/81
E. 1/729

How do u even go about solving this???

Kushal

[Rahul@gurome]

If x is prime and nx is both the square of an integer and the cube of an integer, where n is a positive integer, what is the greatest possible value of (1/nx)?

A. 1
B. 1/32
C. 1/64
D. 1/81
E. 1/729

How do u even go about solving this???

Kushal

The question asks for the greatest possible value of (1/nx).
Which can be rephrased as: What is the least possible value of (nx)?

Now x is a prime and nx is both the square of an integer and the cube of an integer, where n is a positive integer.
Thus the question becomes: What is the smallest positive integer which is both the square of an integer and the cube of an integer?

As x is prime and we have to minimize nx, x should be minimum.
Thus, x must be equal to 2. As nx is simultaneously square of an integer and the cube of an integer, nx must be of the form nx = (a^6)*(b^6)... , where a, b... are primes.

So minimum value of nx = 2^6 = 64.

Largest possible value of (1/nx) = 1/64.

The correct answer is C.

[kushal.adhia]

As nx is simultaneously square of an integer and the cube of an integer, nx must be of the form nx = (a^6)*(b^6)... , where a, b... are primes.

How do u get the 6 in the equation?

Thanks

[Rahul@gurome]

As nx is simultaneously square of an integer and the cube of an integer, nx must be of the form nx = (a^6)*(b^6)... , where a, b... are primes.

How do u get the 6 in the equation?

Thanks

Suppose n is an integer which is simultaneously square of an integer (Say, a) and the cube of an integer (Say, b), then n must be such that, n = a^2 and n = b^3. Thus, a^2 = b^3 => a = b^(3/2) and b = a^(2/3).

As a and b must be integer, b must be square (or any even power) of an integer. Otherwise b^(3/2) cannot be an integer. Also a must be cube (or any multiple power of 3) of an integer, otherwise a^(2/3) cannot be an integer. Therefore, we can write a and b as,

• (1) a = c^3m
  (2) b = d^2m

Where, m is an integer.

Now, n = a^2 = (c^3m)^2 = c^6m
Or in the same way, n = d^6m

Thus for n to be simultaneously square of an integer and the cube of an integer, n must be a power of 6 (or any multiple of 6) of an integer. As the question asks for minimum such value, the power is 6.

Hope this helps.

[Dani@MasterGMAT]

Reverse Plug in the answer choices.
Look at the answer choices: they're all fractions of the form 1/nx. nx is actually the values at the bottom of the fraction - you can now go and find which of the values 1, 32, 64, 81, 729 is both a square and a cube of a number. Since you want the greatest value of the fraction, start from the first answer choice A 1, which is actually a "fraction" 1/1 where nx is 1.
1 is the square and the cube of 1, but remember that x has to be an integer.
B 32 is neither a square nor a cube.
C 64 - is the square of 8 and the cube of 4, so it fits. This will be the greatest value - the remaining answer choices, even if they fit the requirements, are going to give a fraction smaller than 1/64, and are thus irrelevant.

[goyalsau]

1 is the square and the cube of 1, but remember that x has to be an integer.

I think you were suppose to write that x has to be a prime number.

Why 1 is not a prime number ?
AS per the definition of prime number , " Numbers which can not be divisible by any other number ( except them selves )
are all prime numbers.

So why 1 is not prime.

[GMATGuruNY]

1 is the square and the cube of 1, but remember that x has to be an integer.

I think you were suppose to write that x has to be a prime number.

Why 1 is not a prime number ?
AS per the definition of prime number , " Numbers which can not be divisible by any other number ( except them selves )
are all prime numbers.

So why 1 is not prime.

A prime number is a positive integer that has exactly two distinct positive factors, 1 and itself. 1 does not have two distinct positive factors -- the only positive factor of 1 is 1 -- so by definition 1 is not prime. The smallest prime number is 2.

[kushal.adhia]

thank you guys