By which smallest number must the following numbers be divided so that the quotient is a perfect cube? (I)7803          (ii) 107811       (iii) 35721     (iv) 243000

Explanation

By using prime factorization we could find:
(i)`7803`
`7803=3^3times17^2`
As we can see that `17^2` is not cubed.
Hence, to make the quotient a perfect cube we divide it by `17^2=289`, which gives `27` as quotient where, `27` is a perfect cube.

(ii)`107811`
`107811=3^3times3times11^3`
As we can see that `3` is not cubed.
Hence, to make the quotient a perfect cube we divide it by `3`, which gives `35937` as quotient where, `35937` is a perfect cube.

(iii)`35721`
`35721=3^6times7^2`
As we can see that `7^2` is not cubed.
Hence, to make the quotient a perfect cube we divide it by `7^2=49`, which gives `729` as quotient where, `729` is a perfect cube.

(iv)`243000`
`234000=2^3times3^2times3^2times5^3`
As we can see that `3^2` is not cubed.
Hence, to make the quotient a perfect cube we divide it by `3^2=9`, which gives `27000` as quotient where, `27000` is a perfect cube.