Question

Find the smalks t number by which the following numbers should be multiplied so that the product is a perfect cube?​

Answer 1

∴To make a perfect cube we need to multiply the product by 5 × 5 = 25. ∴To make a perfect cube we need to multiply the product by 17.

Answer 2

$\bold{ANSWER≈}$

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

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

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)

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

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

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

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

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

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

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

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