4 is the SMALLEST number in which 16384 can be divided so that the quotient may be a perfect cube.

Given:

16384

To FIND:

Find the smallest number by which 16384 can be divided so that the quotient may be perfect cube.

Solution:

The dividend of the question is 16384

The DIVISOR of the question is X

The property of the quotient is that it is a perfect square.

Thereby, let start by taking out the prime factorization of 16384 which is

2^{3} \times 2^{3} \times 2^{3} \times 2^{3} \times 2^{2}2

3

×2

3

×2

3

×2

3

×2

2

Now as we can see that there are four 2^{3}2

3

which if MULTIPLIED will give a perfect cube but the number multiplied by those four 2^{3}2

3

is 4. 4 is the only number which is not a cube there by if we take out 4 from the factorization then the product of 2^32

3

will be perfect cube. Hence if 16384 is divided by 4, then the quotient remaining is \left(2^{3}\RIGHT)^{4}=4096.(2

3

)

4

=4096.

Therefore, the smallest number that can be divided to 16384 to give the quotient a perfect cube is 4.