Question

find the smallest whole number by which each of the following numbers must be divided to obtain a perfect cube. (1) 81 (2) 704​

Answer 1

Answer:

In the prime factorization of a perfect cube every  prime factor occurs 3 times.

 

To determine whether a number is a perfect cube or not proceed as follows:

1.Find the prime factors of the given number.

2. Make Group of 3 equal prime factors.

3. If a group contains only one or two equal prime factors then a given number is not a perfect cube otherwise it is a perfect cube.

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Solution:

i)

81 = (3 x 3 x 3) x 3

Here, one 3 is left which does not form a triplet.

If we divide 81 by 3 , then it will become a perfect cube.

81/3 =  (3 x 3 x 3) x 3[ns1] /3

27= (3 x 3 x 3) is a perfect cube.

Hence, the smallest number by which 81 should be divided to make it a perfect cube is 3.

ii)

128

128 = (2 x 2 x 2) x (2 x 2 x 2) x 2

Here, one 2 is left which does not form a triplet.

If we divide128 by 2 , then it will become a perfect cube

128/2 =  (2 x 2 x 2) x (2 x 2 x 2) x 2/2

 

64= (2 x 2 x 2) x (2 x 2 x 2) is a perfect cube.

Hence, the smallest number by which 128 should be divided to make it a perfect cube is 2.

(iii) 135

135 = 5 x (3 x 3 x 3)

Here, one 5 is left which does not form a triplet.

If we divide135 by 5 , then it will become a perfect cube

135/5 = 5/5 x (3 x 3 x 3)

 

27= (3 x 3 x 3) is a perfect cube.

Hence, the smallest number by which 135 should be divided to make it a perfect cube is 5.

 

(iv) 192

192 = (2 x 2 x 2) x (2 x 2 x 2) x 3

Here, one 3 is left which does not form a triplet.

If we divide192 by 3 , then it will become a perfect cube

192/3 =   (2 x 2 x 2) x (2 x 2 x 2) x 3/3

64=  (2 x 2 x 2) x (2 x 2 x 2) is a perfect cube.

Hence, the smallest number by which 192 should be divided to make it a perfect cube is 3.

 

(v) 704

704 = (2 x 2 x 2) x (2 x 2 x 2) x 11

Here, one 11 is left which does not form a triplet.

If we divide 704 by 11, then it will become a perfect cube

704/11=  (2 x 2 x 2) x (2 x 2 x 2) x 11/11

64= (2 x 2 x 2) x (2 x 2 x 2)  is a perfect cube.

Hence, the smallest number by which 704 should be divided to make it a perfect cube is 11.

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hope this will help you....

Answer 2

Answer:

1 ) Prime factors of 81 = 3*3*3*3

Therefore 81 must be divided by 3 to make it a perfect cube.

2 ) Prime factors of 704 =2*2*2*2*2*2*11

one factor 11 does not appear in a triplet.

Therefore, 704 must be divided by 11 to make it a perfect cube.

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