What is the smallest number by which 1125 must be multiplied so that the product is a perfect cube?

Solution:

(i) 243

Prime factors of 243 =

Here 3 do not appear in 3’s group.

Therefore, 243 must be multiplied by 3 to make it a perfect cube.

(ii) 256

Prime factors of 256 = 2\times2\times2\times2\times2\times2\times2\times2

Here one factor 2 is required to make a 3’s group.

Therefore, 256 must be multiplied by 2 to make it a perfect cube.

(iii) 72

Prime factors of 72 = 2\times2\times2\times3\times3

Here 3 does not appear in 3’s group.

Therefore, 72 must be multiplied by 3 to make it a perfect cube.

(iv) 675

Prime factors of 675 = 3\times3\times3\times5\times5

Here factor 5 does not appear in 3’s group.

Therefore 675 must be multiplied by 3 to make it a perfect cube.

(v) 100

Prime factors of 100 = 2\times2\times5\times5

Here factor 2 and 5 both do not appear in 3’s group.

Therefore 100 must be multiplied by 2\times5= 10 to make it a perfect cube.

Solution:

A number is a perfect cube only when each factor in the prime factorization of the given number exists in triplets. Using this concept, the smallest number can be identified.

(i) 243

243 = 3 × 3 × 3 × 3 × 3

= 33 × 32

Here, one group of 3's is not existing as a triplet. To make it a triplet, we need to multiply by 3.

Thus, 243 × 3 = 3 × 3 × 3 × 3 × 3 × 3 = 729 is a perfect cube

Hence, the smallest natural number by which 243 should be multiplied to make a perfect cube is 3.

(ii)

256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

= 23 × 23 × 2 × 2

Here, one of the groups of 2’s is not a triplet. To make it a triplet, we need to multiply by 2. 

Thus, 256 × 2 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 512 is a perfect cube

Hence, the smallest natural number by which 256 should be multiplied to make a perfect cube is 2.

(iii) 72

72 = 2 × 2 × 2 × 3 × 3

= 23 × 32

Here, the group of 3’s is not a triplet. To make it a triplet, we need to multiply by 3.

Thus, 72 × 3 = 2 × 2 × 2 × 3 × 3 × 3 = 216 is a perfect cube

Hence, the smallest natural number by which 72 should be multiplied to make a perfect cube is 3.

(iv) 675

675 = 5 × 5 × 3 × 3 × 3

= 52 × 33

Here, the group of 5’s is not a triplet. To make it a triplet, we need to multiply by 5. 

Thus, 675 × 5 = 5 × 5 × 5 × 3 × 3 × 3 = 3375 is a perfect cube

Hence, the smallest natural number by which 675 should be multiplied to make a perfect cube is 5.

(v) 100

100 = 2 × 2 × 5 × 5

= 22 × 52

Here both the prime factors are not triplets. To make them triplets, we need to multiply by one 2 and one 5.

Thus, 100 × 2 × 5 = 2 × 2 × 2 × 5 × 5 × 5 = 1000 is a perfect cube

Hence, the smallest natural number by which 100 should be multiplied to make a perfect cube is 2 × 5 =10

☛ Check: NCERT Solutions for Class 8 Maths Chapter 7

Video Solution:

Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube. (i) 243 (ii) 256 (iii) 72 (iv) 675 (v) 100

NCERT Solutions for Class 8 Maths Chapter 7 Exercise 7.1 Question 2

Summary:

The smallest number by which each of the following numbers must be multiplied to obtain a perfect cube.(i) 243 (ii) 256 (iii) 72 (iv) 675 (v) 100 are (i) 3, (ii) 2, (iii) 3, (iv) 5, and (v) 10

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