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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

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(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

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