What is the least number by which 43200 must be multiplied so that the product becomes a perfect square?
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Please disable adblock in order to continue browsing our website.Unfortunately, in the last year, adblock has now begun disabling almost all images from loading on our site, which has lead to mathwarehouse becoming unusable for adlbock users. 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) 100NCERT 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 ☛ Related Questions:
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