By which smallest number must the following numbers be divided so that the quotient is a perfect, cube?
(i) 675 (ii) 8640
(iii) 1600 (iv) 8788
(v) 7803 (vi) 107811
(vii) 35721 (viii) 243000
(i) 675
Prime factors of 675 = 3 × 3 × 3 × 5 × 5 = 33 × 52
We find that 675 is not a perfect cube.
Hence, for making the quotient a perfect cube we divide it by 52 = 25, which gives 27 as quotient and we know that 27 is a perfect cube .
(ii) 8640
Prime factors of 8640 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 5 = 23 × 23 × 33 × 5
We find that 8640 is not a perfect cube.
Hence, for making the quotient a perfect cube we divide it by 5 , which gives 1728 as quotient and we know that 1728 is a perfect cube.
(iii) 1600
Prime factors of 1600 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 = 23 × 23 × 52
We find that 1600 is not a perfect cube.
Hence, for making the quotient a perfect cube we divide it by 52 = 25, which gives 64 as quotient and we know that 64 is a perfect cube
(iv) 8788
Prime factors of 8788 = 2 × 2 × 13 × 13 × 13 = 22 × 133
We find that 8788 is not a perfect cube.
Hence, for making the quotient a perfect cube we divide it by 4, which gives 2197 as quotient and we know that 2197 is a perfect cube
(v) 7803
Prime factors of 7803 = 3 × 3 × 3 × 17 × 17 = 33 × 172
We find that 7803 is not a perfect cube.
Hence, for making the quotient a perfect cube we divide it by 172 = 289 , which gives 27 as quotient and we know that 27 is a perfect cube
(vi) 107811
Prime factors of 107811 = 3 × 3 × 3 × 3 × 11 × 11 × 11 = 33 × 113 × 3
We find that 107811 is not a perfect cube.
Hence, for making the quotient a perfect cube we divide it by 3, which gives 35937 as quotient and we know that 35937 is a perfect cube.
(vii) 35721
Prime factors of 35721 = 3 × 3 × 3 × 3 × 3 × 3 × 7 × 7 = 33 × 33 × 72
We find that 35721 is not a perfect cube.
Hence, for making the quotient a perfect cube we divide it by 72 = 49, which gives 729 as quotient and we know that 729 is a perfect cube
(viii) 243000
Prime factors of 243000 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 5 × 5 × 5 = 23 × 33 × 53 × 32
We find that 243000 is not a perfect cube.
Hence, for making the quotient a perfect cube we divide it by 32 = 9, which gives 27000 as quotient and we know that 27000 is a perfect cube