What is the smallest number by which 3087 must be multiplied so that product is a perfect cube What will be the cube root of the number obtained?

Solution:

A number is a perfect cube only when each factor in the prime factorization is grouped in triples. Using this concept, the smallest number can be identified.

(i) 81

81 = 3 × 3 × 3 × 3

= 33 × 3

Here, the prime factor 3 is not grouped as a triplet. Hence, we divide 81 by 3, so that the obtained number becomes a perfect cube.

Thus, 81 ÷ 3 = 27 = 33 is a perfect cube.

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

(ii) 128

128 = 2 × 2 × 2 × 2 × 2 × 2 × 2

= 23 × 23 × 2

Here, the prime factor 2 is not grouped as a triplet. Hence, we divide 128 by 2, so that the obtained number becomes a perfect cube.

Thus, 128 ÷ 2 = 64 = 43 is a perfect cube.

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

(iii) 135

135 = 3 × 3 × 3 × 5

= 33 × 5

Here, the prime factor 5 is not a triplet. Hence, we divide 135 by 5, so that the obtained number becomes a perfect cube.

135 ÷ 5 = 27 = 33 is a perfect cube.

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

(iv) 192

192 = 2 × 2 × 2 × 2 × 2 × 2 × 3

= 23 × 23 × 3

Here, the prime factor 3 is not grouped as a triplet. Hence, we divide 192 by 3, so that the obtained number becomes a perfect cube.

192 ÷ 3 = 64 = 43 is a perfect cube

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

(v) 704

704 = 2 × 2 × 2 × 2 × 2 × 2 × 11

= 23 × 23 × 11

Here, the prime factor 11 is not grouped as a triplet. Hence, we divide 704 by 11, so that the obtained number becomes a perfect cube.

Thus, 704 ÷ 11 = 64 = 43 is a perfect cube

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

☛ Check: NCERT Solutions for Class 8 Maths Chapter 7

Video Solution:

Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube (i) 81 (ii) 128 (iii) 135 (iv) 192 (v) 704

NCERT Solutions for Class 8 Maths Chapter 7 Exercise 7.1 Question 3

Summary:

The smallest number by which each of the following numbers must be divided to obtain a perfect cube (i) 81 (ii) 128 (iii) 135 (iv) 192 (v) 704 are (i) 3, (ii) 2, (iii) 5, (iv) 3, and (v) 11

☛ Related Questions:

A perfect cube is a number that can be expressed as the product of an integer three times or as the power of three.

For example consider 1029,

Decomposing 1029 into prime factors, we get

1029 = 3 x 7 x 7 x 7

In the product of prime factors, we see three 7's, but we see only one 3. To group this, we need two more 3's.

1029 = 3 x 7 x 7 x 7 x 3 x 3 ==> 9261

9261 is a perfect cube.

Problem 1 :

What least number is to be multiplied with 3087 in order to get a perfect cube?

Solution :

Prime factorization of 3087 is,

3087 = 3 × 3 × 7 × 7 × 7

= 73 × 3²

Hence, the smallest number by which 3087 must be multiplied to obtain a perfect cube is 3.

Problem 2 :

What is the least number by which 12800 should be divided in order to get a perfect cube?

Solution :

12800 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5

= 29 × 5²

12800 is divided by 25 perfect cube is 512.

So, the cube root is 8.

Problem 3 :

Find the smallest common multiple of 48, 72 and 32 that is a perfect cube.

Solution :

LCM (48, 72, 32) = 2 x 2 x 2 x 3 x 2 x 3 x 2

= 288

Decomposing 288, we get

= 2 x 2 x 2 x 3 x 3 x 2 x 2

To make it as perfect cube, we need one more 2 and one more 3.

So, the smallest number to be multiplied is 6.

Problem 4 :

By which least number should 72000 be multiplied such that the result is a perfect cube?

Solution :

Decomposing 72000, we get

72000 = 72 x 10 x 10 x 10

= 2 x 2 x 2 x 3 x 3 x 10 x 10 x 10

To group the values, we need one more 3.

So, 3 is the least number to be multiplied to make it is perfect cube.

Problem 5 :

By which least number should 171500 be divided such that the result is a perfect cube ?

Solution :

171500 = 5 x 7 x 7 x 7 x 5 x 2 x 5 x 2

= 53 x 73 x 22

So, the smallest number to be multiplied to make it as prefect cube is 2.

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Text Solution

Solution : Given: The prime factor of `3087` are `=>3087=3xx3xx7xx7xx7` Since, `3` does not occur in triplet. So, `3087` is not a perfect cube. In order to make a perfect cube .We have to divided by `9`