Find the smallest natural number that we need to divide 14 with to make it a perfect cube.
|
Solution: Show
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) 704NCERT 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:
(i) We have, 1536 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 After grouping the prime factors in triplets, it’s seen that one factor 3 is left without grouping. 1536 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × 3 So, in order to make it a perfect cube, it must be divided by 3. Thus, the smallest number by which 1536 must be divided to obtain a perfect cube is 3. (ii) We have, 10985 = 5 × 13 × 13 × 13 After grouping the prime factors in triplet, it’s seen that one factor 5 is left without grouping. 10985 = 5 × (13 × 13 × 13) So, it must be divided by 5 in order to get a perfect cube. Thus, the required smallest number is 5. (iii) We have, 28672 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 7 After grouping the prime factors in triplets, it’s seen that one factor 7 is left without grouping. 28672 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × 7 So, it must be divided by 7 in order to get a perfect cube. Thus, the required smallest number is 7. (iv) 13718 = 2 × 19 × 19 × 19 After grouping the prime factors in triplets, it’s seen that one factor 2 is left without grouping. 13718 = 2 × (19 × 19 × 19) So, it must be divided by 2 in order to get a perfect cube. Thus, the required smallest number is 2. How do you find the smallest number to be divided to get a perfect cube?Prime factorising 81, we get,. We know, a perfect cube has multiples of 3 as powers of prime factors.. Here, number of 3's is 4.. So we need to divide the factorization by 3 to make 81 a perfect cube.. Hence, the smallest number by which 81 must be divided to obtain a perfect cube is 3.. What is the smallest number that should be multiplied to make it a perfect cube?Suppose if we have 13 in the form of a cube, then it will be a perfect cube. To make the 1352 as a perfect we have to multiply the number by 13. Therefore 13 is the smallest number which should be multiplied to 1352 to get a perfect cube. So, the correct answer is “ 13”.
Is 53240 is a perfect cube?So, 53240 is not a perfect cube. In the factorisation 5 appears only one time. If we divide the number by 5, then the prime factorisation of the quotient will not contain 5. Hence the smallest number by which 53240 should be divided to make it a perfect cube is 5.
|
