What is the smallest number by which 704 must be divided so that the quotient is a perfect cube?

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Ex 7.1, 3 Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube. (v) 704 We see that 704 = 2 × 2 × 2 × 2 × 2 × 2 × 11 Since 11 does not occur in triplets, ∴ 704 is not a perfect cube. So, we divide by 11 to make triplet So, our number becomes 704 × 𝟏/𝟏𝟏 = 2 × 2 × 2 × 2 × 2 × 2 × 11 × 𝟏/𝟏𝟏 = 2 × 2 × 2 × 2 × 2 × 2 Now, it becomes a perfect cube. So, we divide 704 by 11 to make it a perfect cube

Find the smallest number by which the following number must be divided to obtain a perfect cube: 704 Maths Q&A

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Solution:

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) 704

NCERT 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

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