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:
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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