Q. 55.0( 3 Votes )

Let us calculate

Answer :

Let us recall what a perfect cube is.

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


(i). For 675.


We need to factorize 675. We get,


675 = 3 × 3 × 3 × 5 × 5


Group these factors into three similar integers. We have


675 = (3 × 3 × 3) × 5 × 5


To make it into a perfect cube, multiply both sides by 5.


We have


675 × 5 = (3 × 3 × 3) × 5 × 5 × 5


Now, group these factors into three similar integers.


3375 = (3 × 3 × 3) × (5 × 5 × 5)


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


Thus, smallest positive number that should be multiplied with 675 is 5.


(ii). For 200.


We need to factorize 200. We get,


200 = 2 × 2 × 2 × 5 × 5


Group these factors into three similar integers. We have


200 = (2 × 2 × 2) × 5 × 5


To make it into perfect cube, multiply both sides by 5.


We have


200 × 5 = (2 × 2 × 2) × 5 × 5 × 5


Now, group these factors into three similar integers.


1000 = (2 × 2 × 2) × (5 × 5 × 5)


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


Thus, smallest positive number that should be multiplied with 200 is 5.


(iii). For 108.


We need to factorize 108. We get,


108 = 2 × 2 × 3 × 3 × 3


Group these factors into three similar integers. We have


108 = (3 × 3 × 3) × 2 × 2


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


We have


108 × 2 = (3 × 3 × 3) × 2 × 2 × 2


Now, group these factors into three similar integers.


216 = (3 × 3 × 3) × (2 × 2 × 2)


We are able to group it in three similar integers. This means, 216 is a perfect cube.


Thus, smallest positive number should be multiplied with 108 is 2.


(iv). For 121.


We need to factorize 121. We get,


121 = 11 × 11


We can’t group it into three similar integers. So, multiply both sides by 11.


121 × 11 = 11 × 11 × 11


Now, group these factors into three similar integers.


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 multiplied with 121 is 11.


(v). For 1225.


We need to factorize 1225. We get,


1225 = 5 × 5 × 7 × 7


We can’t group it into three similar integers. So, multiply both sides by (5 × 7).


1225 × 5 × 7 = 5 × 5 × 7 × 7 × (5 × 7)


Now, group these factors into three similar integers.


42875 = (5 × 5 × 5) × (7 × 7 × 7)


Since, we are able to group it in three similar integers. This means, 42875 is a perfect cube.


Thus, smallest positive number that should be multiplied with 42875 is (5 × 7), i.e., 35.


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