# find the smallest number by which 675 must be multiplied to obtain a perfect cube.

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## Find the smallest number by which the following number must be multiplied to obtain a perfect cube: 675

Find the smallest number by which the following number must be multiplied to obtain a perfect cube: 675 - Get the answer to this question and access a vast question bank that is tailored for students.

Academic QuestionsMaths QuestionsFind The Smallest Number By Which The Following Number Must Be Multiplied To Obtain A Perfect Cube 675

## Find the smallest number by which the following number must be multiplied to obtain a perfect cube: 675

We have to find the smallest number by which 675 must be multiplied to obtain a perfect cube

**Solution**

675

On prime factorisation of 675 we get,

675 = 3×3×3×5×5

By assorting the factors in triplets of equal factors, 675 = (3×3×3) x 5×5

Here, 5 cannot be arranged into triplets of equal factors.

∴ We will multiply 675 by 5 to get perfect cube.

**Answer**

The smallest number by which 675 must be multiplied to obtain a perfect cube is 5.

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## What is the smallest number by which 675 must be multiplied so that the product is a perfect cube?

Click here👆to get an answer to your question ✍️ What is the smallest number by which 675 must be multiplied so that the product is a perfect cube?

Question

## What is the smallest number by which 675 must be multiplied so that the product is a perfect cube?

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Updated on : 2022-09-05

Solution Verified by Toppr

Correct option is A)

Prime factorising 675, we get,675=5×5×3×3×3 =3 3 ×5 2 .

We know, a perfect cube has multiples of 3 as powers of prime factors.

Here, number of 3's is 3 and number of 5's is 2.

So we need to multiply another 5 in the factorization to make 675 a perfect cube.

Hence, the smallest number by which 675 must be multiplied to obtain a perfect cube is 5.

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## 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

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. The smallest numbers required."

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

**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) 100

NCERT 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

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Find the cube root of each of the following numbers by prime factorization method. (i) 64 (ii) 512 (iii) 10648 (iv) 27000 (v) 15625 (vi) 13824 (vii) 110592 (viii) 46656 (ix) 175616 (x) 91125

State true or false. (i) Cube of any odd number is even. (ii) A perfect cube does not end with two zeros. (iii) If square of a number ends with 5, then its cube ends with 25. (iv) There is no perfect cube which ends with 8. (v) The cube of a two digit number may be a three digit number. (vi) The cube of a two digit number may have seven or more digits. (vii) The cube of a single digit number may be a single digit number.

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