Q. 65.0( 3 Votes )

Let us calculate by which positive smallest number should be divided with the numbers below so that the quotient will be a perfect cube.

i. 7000 ii. 2662

iii. 4394 iv. 6750

v. 675

Answer :

Let us recall what a perfect cube is.

A perfect cube is a number that is the cube of an integer.


(i). For 7000.


We need to factorize 7000. We get,


7000 = 7 × 2 × 5 × 2 × 5 × 2 × 5


Group these factors into three similar integers.


7000 = (2 × 2 × 2) × (5 × 5 × 5) × 7


To make it into perfect cube, divide it by 7 on both sides.



1000 = (2 × 2 × 2) × (5 × 5 × 5)


Since, we are able to group it in three similar integers. This means, 1000 is a perfect cube.


Thus, smallest positive number that should be divided with 7000 is 7.


(ii). For 2662.


We need to factorize 2662. We get,


2662 = 2 × 11 × 11 × 11


Group these factors into three similar integers.


2662 = (11 × 11 × 11) × 2


To make it into perfect cube, divide it by 2 on both sides.



1331 = (11 × 11 × 11)


Since, we are able to group it in three similar integers. This means, 1331 is a perfect cube.


Thus, smallest positive number that should be divided with 2662 is 2.


(iii). For 4394.


We need to factorize 4394. We get,


4394 = 2 × 13 × 13 × 13


Group these factors into three similar integers.


4394 = (13 × 13 × 13) × 2


To make it into perfect cube, divide it by 2 on both sides.



2197 = (13 × 13 × 13)


Since, we are able to group it in three similar integers. This means, 2197 is a perfect cube.


Thus, smallest positive number that should be divided with 4394 is 2.


(iv). For 6750.


We need to factorize 6750. We get,


6750 = 5 × 5 × 5 × 3 × 3 × 3 × 2


Group these factors into three similar integers.


6750 = (5 × 5 × 5) × (3 × 3 × 3) × 2


To make it into perfect cube, divide it by 2 on both sides.



3375 = (5 × 5 × 5) × (3 × 3 × 3)


Since, we are able to group it in three similar integers. This means, 3375 is a perfect cube.


Thus, smallest positive number that should be divided with 6750 is 2.


(v). For 675.


We need to factorize 675. We get,


675 = 5 × 5 × 3 × 3 × 3


Group these factors into three similar integers.


675 = (3 × 3 × 3) × 5 × 5


To make it into perfect cube, divide it by (5 × 5) on both sides.



27 = (3 × 3 × 3)


Since, we are able to group it in three similar integers. This means, 27 is a perfect cube.


Thus, smallest positive number that should be divided with 675 is (5 × 5), i.e., 25.


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