Find the smallest number by which the number 1296 must be divided to obtain a perfect cube
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Solution Calculate the prime factorisation of 135.135=3×3×3×5To make it a perfect cube, it should be divided by 5 so that the factors can be grouped into triplets. ⇒1355=3×3×3×55=3×3×3 =27which is a perfect cube of 3.Hence, the smallest number by which 135 must be divided to obtain a perfect cube is 5Solution: 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:
What is the perfect cube of 1296?The prime factorization of 1296 is 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3, hence, the cube root of 1296 in its lowest radical form is expressed as 6 ∛6.
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Cube Root of 1296.. 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 by which the number 10584 must be multiplied to obtain a perfect cube?Hence, the smallest number by which 10584 must be multiplied to obtain a perfect cube is 7.
What is the smallest number by which 3000 be divided to make it a perfect cube?Correct Option: A
Clearly, when we divide 3000 by natural number 3, the quotient is 1000 which is a perfect cube.
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